A Gallery of Mathematical Research

Randomly oriented Manhattan lattice

Random walks with memory

The randomly oriented Manhattan lattice

Definition. The randomly oriented Manhattan lattice is the square lattice \(\mathbb Z^2\) with every horizontal and vertical line given a direction by an independent fair coin. At each step the walk chooses one of the two lines through its position with equal probability and moves one unit in that line’s direction.

Once a line is oriented, every later step along it has the same direction.

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One orientation per line · two possible steps from every site

One sampled environment and 20,000 steps of the walk on \(\mathbb Z^2\), with no boundary. Gold arrows mark the two possible steps, each taken with probability 1/2. Follow shows the lattice near the walker; Whole path shows all steps taken so far. The slider sets the number of steps shown; Step adds one step.

Theorem. Record the line orientations by \(\omega\colon\mathbb Z^2\times\{1,2\}\to\{-1,1\}\): the line through \(z\) parallel to \(e_i\) points along \(\omega(z,i)e_i\). Call \(\omega\) the environment, and write \(p_n^\omega(z,z')\) for the probability that the walk started at \(z\) in the environment \(\omega\) is at \(z'\) after \(n\) steps. For almost every environment \(\omega\), \[\sum_{n\geq0}p_n^\omega(x,x)<\infty\qquad\text{for every }x\in\mathbb Z^2.\] That is, the walk is transient almost surely. (Bou-Rabee–Peres, Theorem 1.1)

Deterministic orientations

One fair coin directs each line, so every edge of a line points the same way. Transience does not hold for every deterministic environment: for the alternating orientations \(\omega(z,1)=(-1)^{z_2}\) and \(\omega(z,2)=(-1)^{z_1}\), the walk can be mapped to simple random walk on \(\mathbb Z^2\), as detailed by Beaton and Holmes (2023), who report this as an observation of Guillotin-Plantard.

A. Bou-Rabee and Y. Peres, “The randomly oriented Manhattan lattice in 2D is transient” (2026), arXiv:2609.07621. Manuscript · Lean formalization.

Once-reinforced random walk

Random walks with memory

Once-reinforced random walk

Definition. In once-reinforced random walk on \(\mathbb Z^d\), every edge has weight one until the walk first crosses it and weight \(\beta\ge1\) from then on. At each step the walker moves to one of the \(2d\) neighbours with probability proportional to the weight of the edge it crosses. For \(\beta>1\), the walk favours previously crossed edges. Let \((X_n)_{n\ge0}\) be the walk started at the origin, and write \(\mathbb P_\beta\) and \(\mathbb E_\beta\) for its law and expectation. The range at time \(n\) is \(R_n=\{X_0,\ldots,X_n\}\).

Crossed edgesRecent path & walker

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Sites visited · \(|R_n|\): 1
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One sample of 24,000 steps on \(\mathbb Z^2\), started at the origin, with no boundary. Crossed edges are coloured by the order of their first crossing, in four groups of equal size, paler for later first crossings; the gold trace is the last 32 steps. The view is rescaled to contain the range. At \(\beta=1\), this is simple random walk. When the walk is paused, the four edges at the walker are drawn, with their weights when space permits.

A first crossing changes an edge’s weight from 1 to 4.

Theorem (Range lower bound). For every \(d\ge2\), there is \(c(d)>0\) such that, for all \(\beta\ge1\) and \(n\ge0\), \[\mathbb E_\beta |R_n|\ge c(d)\left(\frac n\beta\right)^{d/(d+1)}.\] In two dimensions the exponent is \(2/3\). (Bou-Rabee–Peres, Theorem 1.1)

Proof overview

Let \(\sigma_k\) be the first time at which the walk has crossed \(k\) distinct edges. The proof shows that \(\mathbb E_\beta\sigma_k\le C(d)\beta k^{1+1/d}\); Markov’s inequality then gives the range lower bound by taking \(k\) of order \((n/\beta)^{d/(d+1)}\). The proof sums exit times of simple random walk over increasing sets, and uses a supermartingale built from the expected time for simple random walk to leave the set of sites whose incident edges have all been crossed. It requires no assumption on the shape of the visited set or on mixing inside it.

The paper also proves a stretched-exponential tail: there is \(C(d)\ge1\) such that, for every \(\beta\ge1\) and every \(n\) with \(n/\beta\ge C(d)\), \(\mathbb P_\beta\{|R_n|<C(d)^{-1}(n/\beta)^{d/(d+1)}\}\le\exp(-C(d)^{-1}(n/\beta)^{(d-1)/(d+1)})\). The Borel–Cantelli lemma then gives \(\liminf_{n\to\infty}|R_n|n^{-d/(d+1)}\ge C(d)^{-1}\beta^{-d/(d+1)}\) almost surely.

A. Bou-Rabee and Y. Peres, “Once-reinforced random walk on \(\mathbb Z^d\) has range exponent at least \(d/(d+1)\)” (2026), arXiv:2610.00090. Paper source · Lean formalization.

Convergence of the random Abelian sandpile

Abelian sandpile

Convergence of the random Abelian sandpile

Abelian sandpiles with random initial states converge almost surely to unique scaling limits.

For each \(x\) in the ball of radius \(R\) in \(\mathbb Z^2\), flip a fair coin. If it lands heads, add 3 grains to \(x\), otherwise add 5 grains; every other site has 0 grains. Then stabilize: a site having at least 4 grains topples by sending one grain to each lattice neighbor.

Initial radius \(R\)

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Sweep: 0
Total grains: 0
Radius: R = 96

Grains per site0123unstable

Each site of the ball of radius \(R\) gets 3 or 5 grains by a fair coin flip; every other site has 0 grains. In each sweep, the unstable sites with \(x_1+x_2\) even topple, and then the unstable sites with \(x_1+x_2\) odd topple. The slider is proportional to the number of topplings so far. All radii use the same coin flips. All views use the coordinates \(x/R\); the odometer view shows \(R^{-2}v_R\), where the odometer \(v_R(x)\) is the number of times \(x\) topples. Two samples of the coin flips are available.

Theorem. Let \(d\ge2\), let \(W\subset\mathbb R^d\) be a bounded Lipschitz domain, and let \(\eta\) be a bounded, stationary, ergodic, integer-valued field satisfying \(\mathbf E\eta(0)>2d-1\). If \(v_n\) is the odometer obtained by putting \(\eta\) on \(nW\cap\mathbb Z^d\) and zero outside, then, with \([x]\) a nearest point of \(\mathbb Z^d\) to \(x\), \(n^{-2}v_n([nx])\) converges almost surely and uniformly to \[\bar v=\min\{v\in C(\mathbb R^d):v\ge0,\ \bar F_\eta(D^2v)\le0\text{ in }W,\ \bar F_0(D^2v)\le0\text{ in }\mathbb R^d\}.\] Here \(\bar F_\eta\) is a nonrandom, degenerate elliptic operator defined implicitly from the law of \(\eta\), \(\bar F_0(M)=\inf\{s\in\mathbb R:\text{some }u:\mathbb Z^d\to\mathbb Z\text{ satisfies }\Delta_{\mathbb Z^d}u\le2d-1\text{ and }u(y)\ge\tfrac12y^{\mathsf T}(M-sI)y\text{ for all }y\in\mathbb Z^d\}\), and the differential inequalities are interpreted in the viscosity sense. The rescaled stable configurations \(\bar s_n(x):=s_n([nx])\) converge weak-* to a deterministic \(s\in L^\infty(\mathbb R^d)\), with \(s=\Delta\bar v+\mathbf E\eta(0)\) weakly in \(W\) and \(s=\Delta\bar v\) weakly outside \(W\). (Bou-Rabee, arXiv:1909.07849, Theorem 1.1)

Theorem. Single source sandpile on a random background. Let \(d\ge2\), let \(\eta_{\min}\in\mathbb Z\), and let \(\eta\) be a stationary ergodic integer-valued field satisfying \(\eta_{\min}\le\eta(x)\le2d-2\) almost surely. Let \(v_n\) be the odometer after adding \(n\delta_0\), and let \([x]\) denote a nearest point of \(\mathbb Z^d\) to \(x\). Almost surely, \(n^{-2/d}v_n([n^{1/d}x])\to\bar v+G\) locally uniformly on \(\mathbb R^d\setminus\{0\}\), where \(G\) is the fundamental solution of the Laplacian in \(\mathbb R^d\), \(\bar F_\eta\) is the deterministic effective operator associated with the law of \(\eta\), and \(\bar v=\inf\{v\in C(\mathbb R^d):v\ge-G,\ \bar F_\eta(D^2(v+G))\le0\}\). The differential inequality is interpreted in the viscosity sense. (Bou-Rabee, arXiv:1909.07849, Theorem 9.2)

Theorem. Exploding sandpiles. Let \(d\ge2\), let \(p\in(0,1]\), and let \(\eta(x)\) be i.i.d., equal to \(2d-1\) with probability \(p\) and \(2d-2\) otherwise. Let \(M_\eta=\min\{m\ge1:\eta+m\delta_0\text{ is not stabilizable}\}\). Add \(M_\eta\delta_0\), use parallel toppling odometers \(v_t\), and set \(T_\eta(x)=\min\{t\ge1:v_t(x)>0\}\). Almost surely, \(n^{-1}T_\eta([nx])\to N_p(x)\) locally uniformly, where \(N_p\) is deterministic, continuous, convex, and one-homogeneous. Moreover, there is a convex domain \(\mathcal B(p)\subset\mathbb R^d\) such that, almost surely, \(|\{\bar v_t>0\}\,\triangle\,\mathcal B(p)|\to0\), where \(\bar v_t(x)=v_t([tx])\). (Bou-Rabee, arXiv:2102.04422, Theorems 1.2–1.3)

Question. Characterize \(\bar F_\eta\), which the first theorem defines only implicitly, in terms of the law of \(\eta\).

Details

Proof idea, first theorem. The sandpile is expressed as the solution to a nonlinear discrete PDE, and the proof follows the stochastic homogenization program of Armstrong and Smart (2014) in three steps: identify the effective equation \(\bar F_\eta\) via the subadditive ergodic theorem applied to an auxiliary Monge–Ampère measure; show that \(\bar F_\eta\) inherits the comparison principle for sandpiles; and show that every subsequential limit of the rescaled odometers is a viscosity solution.

Crossing time. Let \(Q_k=\{x\in\mathbb Z^d:1\le x_i\le k\}\) and let \(L\) be a line crossing \(Q_k\) in a coordinate direction. Run parallel toppling with the odometer frozen at 0 outside \(Q_k\) and at 1 on \(L\). If every line in \(Q_k\) parallel to \(L\) contains at least one site with \(2d-1\) grains, then every site of \(Q_k\) topples within \(k^d\) steps. Under the product measure, this event has probability at least \(1-k^{d-1}e^{-2\epsilon^2k}\) for every \(0<\epsilon<p\).

Proof idea, third theorem. A high-probability bound on the crossing time of a large cube gives a coarsened lattice with an infinite cluster of good cubes; with large-deviation bounds for the chemical distance of supercritical Bernoulli percolation, this gives uniform linear bounds on the arrival times. The arrival times are not subadditive, so the proof passes to the last-wave, an exploding sandpile in which the origin is constrained to topple a fixed number of times. The last-wave is approximated by a quantity which is exactly subadditive, and a deterministic comparison, which requires \(\eta\ge2d-2\), shows that the last-wave approximates the expanding front of the exploding sandpile.

For each \(d\ge2\), arXiv:2102.04422 (Theorem 6.1) also constructs explosive backgrounds \(\eta\not\ge2d-2\) on \(\mathbb Z^d\), random or periodic, which fail to have a limit shape: with \(T(n)=\min\{t>0:v_t(ne_1)>0\}\), \(\limsup_{n}n^{-1}T(n)\ge3/2\) and \(\liminf_{n}n^{-1}T(n)=1\).

A. Bou-Rabee, “Convergence of the random Abelian sandpile”, Ann. Probab. 49 (2021), arXiv:1909.07849; “A shape theorem for exploding sandpiles”, Ann. Appl. Probab. 34 (2024), arXiv:2102.04422. Growth and explosion in sandpiles were studied by A. Fey, L. Levine and Y. Peres, J. Stat. Phys. 138 (2010).

Sandpile avalanches in two dimensions

Abelian sandpile

Sandpile avalanches in two dimensions

Definition. A sandpile on \(\mathbb Z^2\) assigns a number of grains to each site. A site with at least four grains topples, sending one grain to each of its four neighbors, and a sandpile is stable if every site has at most three grains. A box \([-N,N]^2\) with wired boundary conditions is the graph obtained from \(\mathbb Z^2\) by identifying all sites outside the box to a single vertex, the sink, and deleting the loops at the sink; grains sent to the sink disappear. In this box, adding grains at uniformly chosen sites and toppling until the sandpile is stable defines a Markov chain whose stationary distribution is uniform on the recurrent sandpiles: the stable sandpiles that can be obtained from every sandpile by a combination of adding grains and selectively toppling unstable sites. The stationary state \(\mu\) on \(\mathbb Z^2\) is the local limit of these uniform measures as the box grows. The topplings that follow the addition of a grain to a stable sandpile form an avalanche. For a sandpile \(\eta\), let \(\eta+\delta_0\) be the sandpile obtained by adding one grain at the origin; if a finite sequence of topplings takes \(\eta+\delta_0\) to a stable sandpile, then \(\eta+\delta_0\) stabilizes. The radius \(\rho\) of the avalanche is the largest \(\ell^\infty\) distance from the origin of a site that topples, with \(\rho=0\) when no site topples and \(\rho=\infty\) when \(\eta+\delta_0\) does not stabilize.

In two dimensions the origin topples infinitely many times in expectation, and the expected radius of an avalanche is infinite (Bhupatiraju, Hanson and Járai). Nevertheless, avalanches on \(\mathbb Z^2\) are almost surely finite.

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\(\mathscr L\): 0

One grain added at the origin · the avalanche is shown wave by wave

A uniformly chosen recurrent sandpile of the box \([-64,64]^2\) with wired boundary conditions, and its spanning tree under the burning bijection. One grain is added at the origin and the avalanche is shown wave by wave; warm shading shows the number of topplings at each site, and white marks the most recent topplings. White paths form the minimal subtree containing the marked sites (the origin, its four neighbors and \((r,0)\)), blue marks the past of their common ancestor (the sites whose tree paths to the sink pass through it), purple is the path from the common ancestor to the sink, and, when the common ancestor is not the sink, the \(\mathscr L\) green edges are the lattice edges from the minimal subtree to vertices outside this past. Thin lines are tree edges from sites whose past has at least twelve sites.

Theorem. For \(\mu\)-almost every sandpile \(\eta\), the sandpile \(\eta+\delta_0\) stabilizes. Moreover, there are constants \(a>0\) and \(C<\infty\) such that every \(M\geq0\) satisfies \[\mu(\rho>M)\leq C(\log(M+2))^{-a}.\] (Bou-Rabee, Theorem 1.1)

Proof idea. While the origin is unstable, topple it once and then stabilize the other sites, treating the origin as a sink; the sites that topple form a wave. Apart from rare cases, a large avalanche contains a wave that contains every marked site followed immediately by one that misses a marked site. Applying the burning bijection separately to a wave and to the rest of the box bounds the expected number of such pairs by \(\mathbb E[1/\mathscr L]\) for a uniform spanning tree of the box. At each of order \(\log\log r\) scales \(s\), effective-conductance and variance bounds show that, given the earlier scales, \(\mathscr L\) is at least a constant times \(\log s\) with probability at least a universal \(\varepsilon>0\), unless an earlier branch of the tree wanders far; hence \(\mathbb E[1/\mathscr L]\le C(\log r)^{-a}\).

A. Bou-Rabee, “Sandpile avalanches in two dimensions are finite” (2026). Manuscript · Lean formalization.

Dynamic dimensional reduction

Abelian sandpile

Dynamic dimensional reduction

Definition. Fix \(d\ge2\) and \(N\ge1\). The hypercube \(\mathcal C_N^{(d)}\) of side length \(N\) is \(\{x+\mathbf1:x\in\mathbb Z^d,\ |x_i|\le k\}\) if \(N=2k+1\), and \(\{x-k\mathbf1:x\in\mathbb Z^d,\ 1\le x_i\le2k\}\) if \(N=2k\). Start with \(2d\) grains at every site of \(\mathcal C_N^{(d)}\), then iterate the following rule: every site with at least \(2d\) grains topples in parallel, simultaneously giving one grain to each of its \(2d\) neighbors. If a boundary site topples, it loses grains over the edge. The odometer \(v_t^{(d,N)}\) counts the topples at each site up to time \(t\).

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Time \(t\): 0
Sites of the two-dimensional corner toppling at time \(t\): 0
Center slice: disagreements up to time \(t\): 0
Selected slice: disagreements up to time \(t\): 0

Parallel toppling on \(\mathcal C_{68}^{(3)}\) with 6 grains at every site and on \(\mathcal C_{68}^{(2)}\) with 4 grains at every site; each is computed on one corner and reflected. Left: the selected slice of \(\mathcal C_{68}^{(3)}\). Right: \(\mathcal C_{68}^{(2)}\). Cream: sites toppling at time \(t\). Light blue: sites that toppled at time \(t-1\). On the center slice, the sites toppling at time \(t\) are those of \(\mathcal C_{68}^{(2)}\), for every \(t\). A disagreement is a pair \((x,s)\) with \(s\le t\) such that \(x\) topples at time \(s\) in exactly one of the two sandpiles; counts refer to one 34-by-34 corner.

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Same toppling sites in every marked corner

Time \(t\) / \(\tau_{16}\): 0 / 241
Compared sites: 1,024
Compared sites where the odometers differ: 0

Marked corners

Parallel toppling on \(\mathcal C_{64}^{(2)}\) and \(\mathcal C_{32}^{(2)}\), starting with 4 grains at every site. Each outlined 16-by-16 corner of \(\mathcal C_{64}^{(2)}\) is compared site by site with the corresponding corner of \(\mathcal C_{32}^{(2)}\). Cream: sites toppling at time \(t\). Pink: sites where the two differ. The odometers agree on all four corners up to time \(\tau_{16}=241\), the first time at which the odometer of \(\mathcal C_{32}^{(2)}\) is at least 16 at every site of its inner boundary.

Theorem. Let \(d\ge2\), \(N\ge1\), \(M=\lceil N/2\rceil\), and \(\mathcal S_M^{(d)}=\{M\ge x_1\ge\cdots\ge x_d\ge1\}\). Start with \(2d\) grains at every site of \(\mathcal C_N^{(d)}\), topple in parallel, with grains lost over the edge when a boundary site topples, and let \(v_t^{(d,N)}\) be the odometer at time \(t\). Then, for every \(t\ge1\) and \(x\in\mathcal S_M^{(d-1)}\), \[v_t^{(d,N)}(x,1)=v_t^{(d-1,N)}(x),\] and, for \(x\in\mathcal S_M^{(d-1)}\) with every coordinate at least 2, the final odometers satisfy \(v_\infty^{(d,N)}(x,2)=v_\infty^{(d-1,N)}(x)\). (Bou-Rabee, arXiv:2009.05968, Theorem 1.1(1))

Theorem. Let \(d\ge2\), \(N\ge1\), \(M=\lceil N/2\rceil\), and \(\mathcal S_M^{(d)}=\{M\ge x_1\ge\cdots\ge x_d\ge1\}\). For \(1\le j<M\), let \(\partial^\circ\mathcal C_{2j}^{(d)}=\{x\in\mathcal C_{2j}^{(d)}:\exists y\notin\mathcal C_{2j}^{(d)},\ |x-y|=1\}\) and define \(\tau_j=\min\{t\ge1:\min_{x\in\partial^\circ\mathcal C_{2j}^{(d)}}v_t^{(d,2j)}(x)\ge j\}\). Then, for every \(t\le\tau_j\) and \(x\in\mathcal S_M^{(d)}\) with \(x_i>M-j\) for every \(i\), \[v_t^{(d,N)}(x)=v_t^{(d,2j)}\bigl(x-(M-j)\mathbf1\bigr).\] Thus the odometer for the cube of side length \(N\) contains the odometer for the cube of side length \(2j\) up to time \(\tau_j\). (Bou-Rabee, arXiv:2009.05968, Theorem 1.1(2))

Question. For \(k\ge0\) and \(d>k+1\), prove the two identities above for the uniform initial configuration \(s_0\equiv2d+k\).

Details

Proof idea. The proof is an induction on hypercube dimension, side length, and time. An iteration of a technique of Babai and Gorodezky (2007) gives symmetry of the odometer, a derivative comparison result, and a parabolic least action principle. The resulting discrete derivative bounds deteriorate as the initial condition grows; only when every site starts with \(2d\) grains are they strong enough to ensure dimensional reduction. If the parallel toppling process for a \(d_0\)-dimensional sandpile is sufficiently regular, then dimensional reduction holds in all dimensions \(d>d_0\), and this regularity is proved for \(d_0=1\).

Equality of the odometers at every time implies equality of their increments. Thus the sites toppling on the center slice at time \(t\) are exactly the sites toppling in the lower-dimensional sandpile at time \(t\). The paper does not know how to prove dimensional reduction for only the terminal sandpile.

The displayed computation compares \(\mathcal C_{68}^{(3)}\) with \(\mathcal C_{68}^{(2)}\) through all 1,295 time steps, on one 34-by-34 corner of each slice. The number of disagreements, pairs \((x,s)\) at which \(x\) topples at time \(s\) in exactly one of the two sandpiles, is 0 on the center slice and 21,149, 35,066, 60,403 and 140,598 on the slices at offset 1, 2, 3 and 7.

The final sandpiles are not equal: for \(x\in\mathcal S_M^{(d-1)}\) with every coordinate at least 2, \(s_\infty^{(d,N)}(x,1)=s_\infty^{(d-1,N)}(x)+2\) (Corollary 1.1 of the paper). The identity of Theorem 1.1 is for odometers.

A. Bou-Rabee, “Dynamic dimensional reduction in the Abelian sandpile”, Comm. Math. Phys. 390 (2022), 933–958, arXiv:2009.05968.

Pareto peeling

Definition. For a compact set \(A\subset\mathbb R^2\) and a norm \(\varphi\), the Pareto hull is \(P(A)=\{x\in\mathbb R^2:\text{ for every }y\ne x\text{ there is }a\in A\text{ with }\varphi(a-x)<\varphi(a-y)\}\). Set \(E_1(A)=P(A)\) and \(E_{k+1}(A)=P(A\cap\operatorname{int}E_k(A))\).

Unit ball

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Points: 0
Points remaining: 0
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Peel index \(k\): 1current peel

Poisson point process of intensity 30,000 in the unit square. Norm: the norm whose unit ball is the selected polygon. At each step, the points on the boundary of the Pareto hull of the remaining points are removed. Colour: the index of the peel at which a point was removed. Yellow: the points on the boundary of the current Pareto hull. For the diamond and the square, the curves are the level sets \(\bar u=s\max\bar u/8\), \(s=1,\ldots,7\), of the solution \(\bar u\) of the limiting equation with \(f\equiv1\).

Theorem. Let \(\varphi\) be a norm on \(\mathbb R^2\) whose unit ball has at least one boundary facet. Let \(U\subseteq\mathbb R^2\) be bounded, open, Pareto efficient, and compatible with \(\varphi\), and let \(f:U\to(0,\infty)\) be bounded and continuous. Let \(u_n=\sum_{k\ge1}1_{\operatorname{int}E_k(X_{nf})}\) be the height function of Pareto hull peeling of a Poisson point process \(X_{nf}\) of intensity \(nf\) in \(U\). Then, on an event of probability one, \(n^{-1/2}u_n\) converges uniformly on \(\overline U\) to the unique viscosity solution of \[\bar H_\varphi(Du)=f\quad\text{in }U,\qquad u=0\quad\text{on }\partial U.\] Let \(\varphi^*\) be the dual norm and \(\mathcal N^*=\{p:\varphi^*(p)=1,\ \#\partial\varphi^*(p)>1\}\). For each \(p\in\mathcal N^*\), choose unit vectors \(w_p,v_p\) with \(w_p\times v_p>0\) such that \(\operatorname{cone}(\partial\varphi^*(p))=\{cv_p-dw_p:c,d\ge0\}\). Then \[\bar H_\varphi(\xi)=\max\!\left\{0,\sup_{p\in\mathcal N^*}\frac{\langle\xi,v_p\rangle\langle\xi,w_p\rangle}{|v_p\times w_p|}\right\}.\] (Bou-Rabee–Morfe, arXiv:2110.06016, Theorem 1.1)

Details

Domain hypotheses. The set \(U\) is Pareto efficient if \(\operatorname{int}(P(\overline U))=U\), where \(P(\overline U)\) is the Pareto hull of \(\overline U\). A convex cone \(Q\) supports \(U\) at \(x_0\in\partial U\) if \((x_0+Q)\cap\overline U\subseteq\partial U\). For \(p\in\mathcal N^*\), the flat cone is \(Q_p=\{aw_p+bv_p:a,b\ge0\}\). The set \(U\) is compatible with \(\varphi\) if, for each \(x_0\in\partial U\), some convex cone that supports \(U\) at \(x_0\) contains \(Q_p\) for some \(p\in\mathcal N^*\). A norm is polyhedral if its unit ball is a polygon. Every bounded, open, Pareto efficient set is compatible with every polyhedral norm, and convex sets are Pareto efficient.

Proof idea. A characterization of Pareto hulls by the flat cones \(Q_p\), due to Durier and Michelot (1986), shows that the height function is determined by a dynamic programming principle. Locally near a point where \(\bar H_\varphi(D\bar u)>0\), an affine transformation of nondominated sorting describes the behaviour of the height function. The upper and lower half-relaxed limits are a viscosity subsolution and a viscosity supersolution of the limiting equation. A series of growth lemmas shows that \(\bar H_\varphi(D\bar u)>0\) holds in the viscosity sense, and compatibility ensures that \(\bar u=0\) on \(\partial U\).

For \(\ell^1\), the cones determined by the four facets of the unit ball are the coordinate quadrants and \(\bar H_\varphi(\xi)=|\xi_1\xi_2|\). On \(U=(0,1)^2\) with \(f\equiv1\), \[u(x,y)=2\sqrt{\min(x,1-x)\min(y,1-y)},\] so \(\max_{\overline U}u=1\).

For \(\ell^\infty\), the cones determined by the facets of the unit ball are the coordinate quadrants rotated by 45 degrees and \(\bar H_\varphi(\xi)=|\xi_1^2-\xi_2^2|/2\). On the unit square, \(u(x,y)=\sqrt2\min(x,1-x,y,1-y)\), with maximum \(\sqrt2/2\); its level sets are concentric squares.

Finite-sample measurements. For three Poisson clouds with mean number of points 30,000 per polygon, the mean of the number of peels divided by the square root of the number of points is 0.952 for the diamond, 0.680 for the square, and 0.428, 0.319, 0.255, and 0.215 for the regular 6-, 8-, 10-, and 12-gons.

For Poisson clouds with mean number of points \(2\times10^5\), the mean over the points \(x\) of the cloud of \(|k(x)/\sqrt N-u(x)|\), where \(k(x)\) is the index of the peel at which \(x\) is removed and \(N\) is the number of points, is 0.015 for \(\ell^1\) and 0.006 for \(\ell^\infty\). If the unit ball is strictly convex, the Pareto hull is the convex hull, and Calder and Smart (2020) showed that convex hull peeling converges to curvature flow, with the larger \(n^{2/3}\) rescaling.

A. Bou-Rabee and P. S. Morfe, “Hamilton-Jacobi scaling limits of Pareto peeling in 2D”, Probab. Theory Related Fields (2024), arXiv:2110.06016. For the \(\ell^1\) norm, this confirms a conjecture of J. Calder. For strictly convex unit balls, convex hull peeling: J. Calder and C. K. Smart, “The limit shape of convex hull peeling”, arXiv:1805.08278. Nondominated sorting of random points has a scaling limit described by an explicit Hamilton–Jacobi equation: J. Calder, S. Esedoǧlu and A. Hero, SIAM J. Math. Anal. 46 (2014).

Integer superharmonic matrices on the F-lattice

Abelian sandpile

Integer superharmonic matrices on the F-lattice

Definition. The F-lattice is the directed periodic planar graph \((\mathbb Z^2,E)\) with \((x\pm e_1,x)\in E\) if \(x_1+x_2\) is even and \((x\pm e_2,x)\in E\) otherwise. Every vertex has two outgoing edges: \(x\to x\pm e_2\) if \(x_1+x_2\) is even and \(x\to x\pm e_1\) if \(x_1+x_2\) is odd. When a vertex has at least as many grains as outgoing edges, it topples, sending one grain along each outgoing edge; so a stable vertex has 0 or 1 grains.

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Local sites

Grains: 1,300,000
View width: 132 sites

Grains per site

The final configuration of the single-source sandpile on the F-lattice: 1,300,000 grains started at the origin. Black: 0 grains. Blue: 1 grain. The view starts 132 sites across and widens to the whole configuration, 1,481 sites across.

Definition. Define \(\Delta_Fg(x)=\sum_{(y,x)\in E}(g(y)-g(x))\). Thus \(\Delta_F\) is the horizontal second difference when \(x_1+x_2\) is even and the vertical second difference when \(x_1+x_2\) is odd. A symmetric matrix \(A\) is integer superharmonic when there is an integer-valued \(g\) on the F-lattice such that \(\Delta_Fg\le0\) and \(g(x)=\tfrac12x^{\mathsf T}Ax+o(|x|^2)\).

Sections at c = −0.92 of the cones {A : A ≼ M(λ₁, λ₂, 0)}, λ ∈ Λ, whose bases form an overlapping circle packing; one intersection chord is marked. Below: the hyperbola from 1/1 to 0/1 and the Laplacians of standard tile odometers at 1/1, 2/3, 1/2, 1/3, 1/5 and 0/1.

Sections of the cones {A : A ≼ M(λ₁, λ₂, 0)}, λ ∈ Λ, whose union is Γ_F; viewed from above, their bases form an overlapping circle packing. M(a, b, c) is the symmetric matrix with diagonal entries (c+a)/2, (c−a)/2 and off-diagonal entry b/2. The cone with apex M(λ₁, λ₂, 0) meets the plane c = −r in the disk of radius r centred at (λ₁, λ₂). Shown: r = 0.92, apexes at the lattice points (s−t, s+t) for integers s, t. Nearest apexes are √2 apart, so the disks overlap. Below: the Laplacians of the standard tile odometers at the rational points 1/1, 2/3, 1/2, 1/3, 1/5 and 0/1 of the hyperbola; extended to ℤ², the odometers have periodic Laplacians.

Recursion. A reduced fraction \(n/d\) is even if \(n+d\) is even and odd otherwise. An odd reduced fraction \(p=n_p/d_p\) and an even reduced fraction \(q=n_q/d_q\) produce an odd-even child pair by \[C(p,q)=\left(\frac{n_p+n_q}{d_p+d_q},\frac{2n_p+n_q}{2d_p+d_q}\right).\] Select a node to compare the two parent tiles with the two child tiles.

Theorem. For directed edges \(E\) of the F-lattice, put \(\Delta_Fg(x)=\sum_{(y,x)\in E}(g(y)-g(x))\). Write \(A\preccurlyeq B\) when \(B-A\) is positive semidefinite, and let \(\Gamma_F\) be the set of \(A\in\mathbb S^2\) for which some \(g:\mathbb Z^2\to\mathbb Z\) satisfies \(\Delta_Fg\le0\) and \(g(x)=\tfrac12x^{\mathsf T}Ax+o(|x|^2)\). Define \(M(a,b,c)=\tfrac12\bigl(\begin{smallmatrix}c+a&b\\b&c-a\end{smallmatrix}\bigr)\) and \(\Lambda=\{(s-t,s+t):s,t\in\mathbb Z\}\). Then \[\Gamma_F=\bigcup_{\lambda\in\Lambda}\{A:A\preccurlyeq M(\lambda_1,\lambda_2,0)\}.\] In \((a,b,c)\)-coordinates the boundary of \(\Gamma_F\) is the graph \[c=-\operatorname{dist}((a,b),\Lambda).\] (Bou-Rabee, arXiv:2110.07556, Theorem 1.1)

Question. Describe the set of integer superharmonic matrices for other periodic planar graphs.

Details

Write \((y,x)\in E\) for the directed edge \(y\to x\). In the paper’s convention, \((x\pm e_1,x)\in E\) when \(x_1+x_2\) is even and \((x\pm e_2,x)\in E\) otherwise, with \(\Delta g(x)=\sum_{(y,x)\in E}(g(y)-g(x))\). Thus \(\Delta\) sums over incoming edges: at a vertex with \(x_1+x_2\) even it uses the horizontal neighbours, whereas the outgoing edges are vertical.

Proof idea. The matrices \(M(s-t,s+t,0)\) lie on \(\partial\Gamma_F\), so by the downwards closure of \(\Gamma_F\) it suffices to show that the intersection curve of each pair of overlapping cones is in \(\partial\Gamma_F\); by symmetry it suffices to check one hyperbola and, since \(\Gamma_F\) is closed, its rational points. A modified Farey recursion constructs a distinct pair of recurrent functions, a standard and an alternate odometer, for each rational point on the hyperbola. The label \(n/d\) corresponds to \(t=(d^2-n^2)/(d^2+2dn-n^2)\). The rotation \(\mathcal R(n,d)=(d-n,d+n)\), with both entries halved when \(n+d\) is even, pairs the displayed points 1/2 with 1/3 and 2/3 with 1/5; it is also a rotation of the lattice vectors of the odometers.

A. Bou-Rabee, “Integer superharmonic matrices on the F-lattice”, Adv. Math. 436 (2024), arXiv:2110.07556. The overlapping circle packing is a Kleinian bug, in the sense of M. Kapovich and A. Kontorovich, arXiv:2104.13838. Pattern formation on this lattice was studied by D. Dhar, T. Sadhu and S. Chandra, EPL 85 (2009), arXiv:0808.1732. See also Wesley Pegden’s sandpile simulation gallery.

Internal DLA on mated-CRT maps

Internal DLA and divisible sandpile

Internal DLA on mated-CRT maps

Definition. Fix \(\gamma\in(0,2)\) and \(\varepsilon>0\). Let \((L_t,R_t)_{t\in\mathbb R}\) be a two-dimensional Brownian motion started at the origin with correlation coefficient \(-\cos(\pi\gamma^2/4)\). The \(\gamma\)-mated-CRT map with cell size \(\varepsilon\) is the graph with vertex set \(\varepsilon\mathbb Z\), with two vertices \(a_1<a_2\) connected by an edge if and only if, for \(Z=L\) or \(Z=R\), \[\max\!\left\{\inf_{[a_1-\varepsilon,a_1]}Z,\inf_{[a_2-\varepsilon,a_2]}Z\right\}\le\inf_{[a_1,a_2-\varepsilon]}Z.\] There are two edges connecting \(a_1\) and \(a_2\) if \(a_2-a_1>\varepsilon\) and the condition holds for both \(Z=L\) and \(Z=R\). In IDLA, each explorer starts at 0 and travels according to a simple random walk until it reaches an unoccupied vertex, at which point it stops and occupies that vertex. In the divisible sandpile, we start with mass \(t\varepsilon^{-1}\) at 0 and zero elsewhere; at each time step, every vertex \(a\) with mass \(\sigma(a)>1\) is unstable and topples, distributing the excess mass \(\sigma(a)-1\) equally among the neighbors of \(a\) (parallel edges counted separately).

\(\gamma=\sqrt2\) · upper edges from \(L\) · lower edges from \(R\) · two edges when the adjacency condition holds for both

Toppling rule. On a locally finite multigraph, a vertex of mass \(m(v)>1\) keeps mass 1 and sends \((m(v)-1)/\deg(v)\) along each incident edge, where parallel edges are counted in \(\deg(v)\). Here \(m(v)=2.5\) and \(\deg(v)=6\), so each of the six edges carries 0.25. Total mass is conserved: \(2.5+6(0.50)=1+6(0.75)=5.5\).

Initial state. Assign a non-negative mass to every vertex.

Unstable vertex. The centre has mass \(2.5>1\).

Toppling. It retains 1 and divides its excess equally among its incident edges.

Abelian property. This step conserves total mass. With a finite amount of mass on an infinite graph, the end configuration exists and does not depend on the order in which unstable sites are toppled.

Setup

LQG harmonic balls

Setup. Fix \(\gamma\in(0,2)\). Let \(h\) be the field of a \(\gamma\)-quantum cone, let \(\mu_h\) be its LQG area measure, and let \(G^\varepsilon\) be the \(\varepsilon\)-mated-CRT map in its SLE/LQG embedding. Write \(H_a^\varepsilon\) for the cell corresponding to \(a\in\varepsilon\mathbb Z\). Let \(\{\Lambda_t\}_{t>0}\) be the growing family of \(\gamma\)-LQG harmonic balls centered at 0 with \(\mu_h(\Lambda_t)=t\); in particular, \[\frac1t\int_{\Lambda_t}f\,d\mu_h=f(0)\] for every function \(f\) harmonic on a neighbourhood of \(\overline{\Lambda_t}\). For \(\delta>0\), define \[B_\delta^+(X)=\{x:\operatorname{dist}(x,X)\le\delta\},\qquad B_\delta^-(X)=\{x\in X:\operatorname{dist}(x,X^c)\ge\delta\}.\]

Theorem (divisible sandpile). Let \(D^\varepsilon(t\varepsilon^{-1})\) be the divisible sandpile cluster started with mass \(t\varepsilon^{-1}\) at the origin, the final collection of vertices which contain some mass, and set \(\mathcal D^\varepsilon(t\varepsilon^{-1})=\bigcup_{a\in D^\varepsilon(t\varepsilon^{-1})}H_a^\varepsilon\). For every \(t,\delta>0\), with probability tending to one as \(\varepsilon\to0\), \[B_\delta^-(\Lambda_t)\subset\mathcal D^\varepsilon(t\varepsilon^{-1})\subset B_\delta^+(\Lambda_t).\] (Bou-Rabee–Gwynne, Theorem 1.6)

Pace

Run

Loading map

Time steps: 0
Positive-mass cells (current / final): 0
Toppled / positive-mass cells: —
Area of the cells of positive-mass vertices / disk area: —
Total excess mass \(\sum_v(m(v)-1)^+\) over interior vertices: —

Dim: positive mass · solid: toppled

Divisible sandpile on a finite \(\gamma=\sqrt2\) mated-CRT map with boundary, with 10,000 vertices, under the Tutte embedding: boundary vertices are placed by discrete harmonic measure and each interior vertex at the average of the positions of its neighbors. Mass 2,000 is placed at a source vertex at maximum graph distance from the boundary. At each time step every unstable vertex topples; parallel edges are counted separately. Solid colour: vertices that have toppled. Dim colour: vertices holding positive mass that have not toppled. Yellow outline: vertices that newly contain mass.

Theorem (IDLA). Let \(A^\varepsilon(0)=\varnothing\). For \(n\ge1\), obtain \(A^\varepsilon(n)\) by starting an independent simple random walk from 0 and adjoining the first vertex it visits outside \(A^\varepsilon(n-1)\). Set \(\mathcal A^\varepsilon(n)=\bigcup_{a\in A^\varepsilon(n)}H_a^\varepsilon\). For every \(t,\delta>0\), with probability tending to one as \(\varepsilon\to0\), \[B_\delta^-(\Lambda_t)\subset\mathcal A^\varepsilon(\lfloor t\varepsilon^{-1}\rfloor)\subset B_\delta^+(\Lambda_t).\] (Bou-Rabee–Gwynne, Theorem 1.4)

Run

Loading map

Settled explorers: 0
Random-walk steps: 0
Maximum graph distance from the source over occupied vertices: 0

Settlement order

Internal DLA on a finite \(\gamma=\sqrt2\) mated-CRT map with boundary, with 10,000 vertices, under the Tutte embedding. Each explorer starts at the source, follows an incident edge chosen uniformly at each step (parallel edges counted separately), and settles at its first unoccupied vertex. Explorers are added until the first time one of them hits a boundary vertex; 3,087 explorers settle before that time. Cells appear in settlement order; the step count includes the boundary-hitting walk. The main view starts near the source and zooms out as the cluster grows. The inset shows the whole disk, with the visible region outlined in yellow. Yellow outlines in the main view mark recent settlements; cells without recorded polygons appear as point markers.

Theorem. Fix \(\gamma\in(0,2)\), set \(Q=2/\gamma+\gamma/2\), and let \(h=h^{\mathbb C}-\alpha_0\log|\cdot|\), where \(h^{\mathbb C}\) is a whole-plane GFF normalized to have zero circle average on \(\partial B_1(0)\), and \(\alpha_0<Q\). Let \(\mu_h\) be the associated \(\gamma\)-LQG area measure. A \(\gamma\)-LQG harmonic ball \(\Lambda_t(z)\) centered at \(z\) has \(\mu_h(\Lambda_t(z))=t\) and satisfies \[\frac1t\int_{\Lambda_t(z)}f\,d\mu_h=f(z)\] for every \(f=f_1+f_2\), where \(f_1\) is harmonic on a neighbourhood of \(\overline{\Lambda_t(z)}\) and \(f_2(x)=\int_O G_O(x,y)\,\nu(dy)\) for some bounded open set \(O\) with \(\Lambda_t(z)\subset\overline O\) and a signed Radon measure \(\nu\) supported in \(O\setminus\Lambda_t(z)\). Almost surely, for every \(z\in\mathbb C\) there is a unique family \((\Lambda_t(z))_{t>0}\) in this class such that \(\mu_h(\partial\Lambda_t(z))=0\), each \(\Lambda_t(z)\) is bounded, connected, contains \(z\), and equals the interior of its closure; the family increases continuously in the Hausdorff topology and \(\bigcap_{t>0}\Lambda_t(z)=\{z\}\). Almost surely, for Lebesgue-almost every \(t>0\), \(\Lambda_t(0)\) is neither a Lipschitz domain nor an LQG metric ball; for every \(t>0\), the boundaries of the connected components of \(\mathbb C\setminus\overline{\Lambda_t(0)}\) are Jordan curves. (Bou-Rabee–Gwynne, arXiv:2208.11795, Theorems 1.1–1.2)

Question. Show that the scaling limit of IDLA and the divisible sandpile on random planar maps other than mated-CRT maps is also described by \(\gamma\)-LQG harmonic balls; for example, on a uniform random planar map, establish convergence of IDLA to a \(\sqrt{8/3}\)-LQG harmonic ball. Provide bounds on the rate of convergence of IDLA on the mated-CRT map, and determine whether its fluctuations around the limit shape have a scaling limit.

Details

Proof idea. The divisible sandpile odometer satisfies the least action principle. Convergence of the discrete Dirichlet problem and of the discrete Green's kernel on the mated-CRT map shows that it converges to its continuum counterpart, the odometer for Hele-Shaw flow; this gives the inner inclusion for the divisible sandpile cluster. Following Lawler, Bramson and Griffeath, a lower bound on the divisible sandpile cluster implies the same lower bound for the IDLA cluster. By the abelian property, the IDLA cluster can be built by first generating the stopped IDLA cluster and then restarting the stopped random walkers; weak harmonic measure estimates and a modification of the iterative argument of Duminil-Copin, Lucas, Yadin and Yehudayoff show that the restarted random walks cannot travel further than distance \(\delta\) from \(\Lambda_t\).

Toppling every unstable vertex at once is a projected Jacobi iteration for the obstacle problem, and the end configuration does not depend on the order in which unstable sites are toppled.

The cell sizes in the mated-CRT map have a multifractal behavior, so equal numbers of vertices need not occupy comparable Euclidean areas. On \(\mathbb{Z}^{d}\), the domain of fully occupied sites of the divisible sandpile formed from a pile of mass \(m\) at the origin satisfies \(B_{r-c}\subset D_m\subset B_{r+c'}\) with \(r=(m/\omega_d)^{1/d}\): the error in the radius is a constant independent of \(m\); with \(k\) sources it converges to a quadrature domain whose planar boundary lies on an algebraic curve of degree \(2k\) (Levine–Peres). The random-map limit is instead an LQG harmonic ball.

A. Bou-Rabee and E. Gwynne, “Internal DLA on mated-CRT maps”, Ann. Probab. 52 (2024), arXiv:2211.04891; and “Harmonic balls in Liouville quantum gravity” (2022), arXiv:2208.11795. The mated-CRT map is from B. Duplantier, J. Miller and S. Sheffield, “Liouville quantum gravity as a mating of trees”, arXiv:1409.7055; its Tutte embedding is treated in E. Gwynne, J. Miller and S. Sheffield, “The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity”, Ann. Probab. 49 (2021), arXiv:1705.11161. For the divisible sandpile on \(\mathbb{Z}^{d}\), L. Levine and Y. Peres, Potential Anal. 30 (2009), arXiv:0704.0688, and J. Analyse Math. 111 (2010), arXiv:0712.3378. D. Zidarov and L. Levine introduced the model independently.

Divisible sandpiles via random walks in random scenery

Internal DLA and divisible sandpile

Divisible sandpiles via random walks in random scenery

Definition. Let \(G\) be infinite, connected, locally finite, and of bounded degree, and let \(\sigma(v)\) be i.i.d. initial masses. A vertex \(v\) with \(\sigma(v)>1\) is unstable; it topples by keeping mass 1 and distributing the excess \(\sigma(v)-1\) equally among its \(\deg(v)\) neighbours. In the parallel toppling procedure all unstable vertices topple simultaneously. The odometer \(u_n(v)\) is the mass sent from \(v\) to each neighbour during the first \(n\) steps, so \(\deg(v)u_n(v)\) is the total mass emitted by \(v\) up to time \(n\). Since \(u_n(v)\) is nondecreasing in \(n\), the limit \(u_\infty(v)=\lim_{n\to\infty}u_n(v)\in[0,\infty]\) exists. If \(u_\infty(v)<\infty\) for every \(v\), the configuration stabilizes; otherwise it explodes. Writing \(\zeta(v)=(\sigma(v)-1)/\deg(v)\) for the per-neighbour excess and \((Pu)(v)\) for the average of \(u\) over the neighbours of \(v\), the odometer satisfies \(u_{n+1}=(\zeta+Pu_n)^+\).

Constant \(a\) in \(g=a+\xi\)

Run

Initial scenery

\(\min_D q_n\) … \(\max_D q_n\): 0
\(\max_{x\in D}w_{n+1}(x)\): 0.000

Per-neighbour excess \(g\)

Odometer recursion of the parallel divisible sandpile on the displayed finite subgraph D of Z², with the walk killed on leaving D: w₀ = 0, qₙ = g + Pwₙ, wₙ₊₁ = max(qₙ, 0), where (Ph)(x) = ¼ Σ h(y) over the neighbours y of x in D, and g = (σ − 1)/4 = a + ξ for a fixed field ξ with values in [−0.58, 0.58) and the selectable constant a. Blue: qₙ > 0. Pink: qₙ < 0. Disc radius increases with |qₙ|. Ring: the set {qₙ > 0}. Yellow: a random walk X started at a maximizer of w₇₂ and stopped at the first k with w₇₂₋ₖ(Xₖ) = 0, or on leaving D. New resamples the walk; g is unchanged.

Theorem. Let \(G\) be infinite, locally finite, connected, and of bounded degree, and let \((\sigma(v))\) be i.i.d. with mean \(\mu\in(-\infty,\infty]\). If \(\mu>1\), then \(\mathbf P(\sigma\text{ stabilizes})=0\). If \(\mu=1\) and either \(0<\operatorname{Var}(\sigma(v))<\infty\), or \(\sigma(v)\not\equiv1\) and \(\sigma(v)-1\) is symmetric, then \(\mathbf P(\sigma\text{ stabilizes})=0\). (Bou-Rabee–Peres–Sava-Huss, arXiv:2604.13968, Theorem 1.2)

Theorem. If \(\mu\in[-\infty,1)\) and \(\mathbf E[(\sigma(v)^+)^p]<\infty\) for some \(p>3\), where \(a^+=\max\{a,0\}\), then \(\sup_v\mathbf E[u_\infty(v)^q]<\infty\) for every \(q\in[1,(p-1)/2)\); in particular \(\mathbf P(\sigma\text{ stabilizes})=1\). (Bou-Rabee–Peres–Sava-Huss, arXiv:2604.13968, Theorem 1.3)

Details

Proof idea. Put \(\zeta(v)=(\sigma(v)-1)/\deg(v)\), let \(X\) be simple random walk, and write \(S_n=\sum_{k=0}^{n-1}\zeta(X_k)\). With \([a]_+=\max\{a,0\}\), and the supremum over stopping times \(\tau\) of \(X\) bounded by \(n\), the parallel odometer satisfies

\[u_{n+1}(x)=\left[\zeta(x)+\frac1{\deg(x)}\sum_{y\sim x}u_n(y)\right]_{+},\qquad u_n(x)=\sup_{\tau\le n}\mathbf E_x[S_\tau\mid\sigma].\]

Thus stabilization is equivalent to finiteness of an optimal-stopping value for random walk in random scenery. For \(\mu<1\), the payoff has negative drift; local-time estimates and a Fuk–Nagaev inequality control the centered fluctuations when \(p>3\). For \(\mu>1\), positive drift makes the value diverge. At \(\mu=1\), if the Green function is not square summable, the variance of the scenery sum grows without bound and a second-moment argument gives explosion; otherwise the walk is stopped in rare regions where the initial masses are unusually small, and an Efron–Stein variance bound upgrades this to quenched explosion. When \(\sigma(v)-1\) is symmetric, a convexity argument reduces the problem to the finite-variance case. This avoids conservation of mean mass, which fails on general graphs.

The moment condition is nearly sharp: for every \(p<3\), the paper constructs bounded-degree graphs on which, for every \(\mu<1\), there exist i.i.d. masses bounded from below with mean \(\mu\) and finite \(p\)-th moment that explode almost surely. It also constructs a locally finite tree on which every i.i.d. law with \(\mathbf E|\sigma(v)|<\infty\) stabilizes almost surely, including laws with \(\mu>1\).

A. Bou-Rabee, Y. Peres and E. Sava-Huss, “Divisible sandpiles via random walks in random scenery” (2026), arXiv:2604.13968.

Sharpness and critical scaling of parking

Internal DLA and divisible sandpile

Sharpness and critical scaling of parking

Definition. Independently at each \(x\in\mathbb Z^d\), place one car with probability \(p\) and one parking spot with probability \(1-p\). In each discrete round, every unparked car takes one step of an independent simple random walk. A car parks when it reaches a spot that is still free; when several cars arrive in the same round at a free spot, one of them, chosen uniformly at random, parks there. The odometer \(U_n(x)\) counts the steps taken from \(x\) during the first \(n\) rounds.

Car density

Run

Cars and free spots

Round: 0
Cars still driving: 0
Steps taken from the origin \(U_n(0)\): 0

Statefree spot parked driving

Parking process on a 32-by-32 discrete torus, with \(\lfloor 1024p\rfloor\) cars at uniformly chosen distinct sites and a free spot at every other site. A moving car is drawn continuously along its nearest-neighbor step. Cell color records \(\log(1+U_n(x))\) on a fixed scale. After all cars park, the same run repeats. Replay keeps the initial configuration and the step sequence of every car; New resamples both.

Setup. Let \(S_t\) be the probability that the origin initially contains a car which is still unparked after round \(t\).

Theorem. At \(p=1/2\), \[\mathbf E U_n(0)\asymp\begin{cases}n^{(4-d)/4},&d\le3,\\ \log n,&d\ge4,\end{cases}\qquad S_t\asymp(t+1)^{-d/4}\quad(d\le3).\] For \(d\ge4\), there are \(0<c\le C<\infty\) such that, for every \(t\ge0\), \(c/(t+1)\le S_t\le C\log(t+2)/(t+1)\).

Theorem. If \(0<p<1/2\), there are \(0<c\le C<\infty\) such that for every \(t\ge1\), \[C^{-1}e^{-Ct^{d/(d+2)}}\le S_t\le Ce^{-ct^{d/(d+2)}}.\] Moreover, with \(U_\infty(0)=\lim_{n\to\infty}U_n(0)\), as \(p\uparrow1/2\), \[\mathbf E U_\infty(0)\asymp\begin{cases}(1-2p)^{-3},&d=1,\\(1-2p)^{-1},&d=2,\\(1-2p)^{-1/3},&d=3,\\\log\!\bigl(e/(1-2p)\bigr),&d\ge4.\end{cases}\]

Proof mechanism

Write \(\eta(x)=1\) at a car and \(\eta(x)=-1\) at a spot. Attach to each site an independent sequence of uniformly chosen neighbors, the destinations of its successive steps, and let \(I_{y,x}(m)\) count the entries equal to \(x\) among the first \(m\) entries at \(y\). Then \(U_0=0\) and \[U_{n+1}(x)=\Bigl(\eta(x)+\sum_y I_{y,x}(U_n(y))\Bigr)^+.\] Replacing each \(I_{y,x}(m)\) by its expected value \(P(y,x)m\), where \(P\) is the transition matrix of simple random walk, gives the divisible sandpile odometer \(u_0=0\), \(u_{n+1}=(\eta+Pu_n)^+\), with \[u_n(x)=\sup_{\tau\le n}\mathbf E_x\sum_{j<\tau}\eta(X_j),\] where \(X\) is simple random walk started at \(x\) and the supremum is over stopping times of \(X\) bounded by \(n\). Jensen's inequality gives \(u_n\le\mathbf E[U_n\mid\eta]\). Revealing each instruction when a particle first uses it expresses the accumulated difference between the arrivals and their means as a sum of bounded martingale differences, and the odometer itself controls the total variance. With a separate lower bound \(\mathbf EU_n(0)\ge c\log n-C\), this shows that \(\mathbf EU_n(0)\) is comparable to \(\mathbf Eu_n(0)+\log n\); below dimension four the same estimates transfer the scaling limit of \(u_n\) to \(U_n\). The subcritical tail is proved without the divisible sandpile: tag a particle at the origin and tilt the law of \(\eta(0)\) to raise the density of particles; a covariance bound by three times the derivative of the survival probability in the tilt parameter, integrated and combined with the Donsker–Varadhan estimate for the range, gives the bounds.

A. Bou-Rabee and C. Panagiotis, “Sharpness and critical scaling of parking” (2026), arXiv:2609.02820.

Quantitative explosion and percolation of the divisible sandpile

Internal DLA and divisible sandpile

Quantitative explosion and percolation of the divisible sandpile

Definition. Let \((\sigma(x))_{x\in\mathbb Z^d}\) be i.i.d. initial masses. A site is unstable when its mass exceeds one, and an unstable site topples, keeping one unit of mass and sharing the excess equally among its \(2d\) neighbors. At each discrete step, every unstable site topples simultaneously. If \(u_t(x)\) is the mass sent from \(x\) to each neighbor through time \(t\), then \[u_0\equiv0,\qquad u_{t+1}(x)=\left(\frac{\sigma(x)-1}{2d}+\frac1{2d}\sum_{y\sim x}u_t(y)\right)^+.\] The function \(u_t(x)\) increases to the odometer \(u_\infty(x)\in[0,\infty]\). For an i.i.d. mass field \(\sigma^{(\rho)}\) with mean \(\rho\), write \(u_\infty^{(\rho)}\) for its odometer; the toppled set is \(\mathcal T^{(\rho)}=\{x:u_\infty^{(\rho)}(x)>0\}\).

Poisson mean

Run

Initial Poisson masses

Time \(t\): 0
Toppled sites: 0
Sites in the largest component: 0

Displayed setother components largest component

Parallel divisible sandpile on an 88-by-88 discrete torus. Initial masses use a monotone coupling of Poisson variables; at \(\rho=1\), the same constant is added to every mass so that the empirical mean is exactly one. For \(\rho<1\), the cells show \(\{u_t>0\}\); for \(\rho=1\), they show \(\{u_t>0.075\sqrt t\}\). Cyan is the largest nearest-neighbor component. Each run displays times \(0\) through \(900\) and then repeats. Brightness increases with \(u_t(x)\); pale yellow briefly marks sites entering the displayed set. The plot below shows the fraction of sites in the displayed set (purple) and in its largest component (cyan) against \(t\), on a \(\log(1+t)\) scale; later times are shown at a higher rate.

Setup. Let \((\mu_\rho)_{\rho\in(0,1]}\) be a family of laws with mean \(\rho\), and let the masses \(\sigma^{(\rho)}(x)\) be i.i.d. with law \(\mu_\rho\). Suppose there are \(\rho_0\in(0,1)\), \(\nu_0>0\), \(\theta_0>0\) and \(K_0<\infty\) such that, for every \(\rho\in[\rho_0,1)\), \(\operatorname{Var}(\sigma^{(\rho)}(0))\ge\nu_0^2\) and \(\mathbf Ee^{\theta_0|\sigma^{(\rho)}(0)-\rho|}\le K_0\).

Theorem (percolation of the toppled set below criticality). For every \(d\ge2\), there exists \(\rho_+\in[\rho_0,1)\), depending on \(d\) and \((\mu_\rho)\), such that, for every \(\rho\in(\rho_+,1)\), the toppled set \(\mathcal T^{(\rho)}\) contains an infinite nearest-neighbor component almost surely.

Theorem (percolation of critical level sets). Let \(d\ge2\) and let \((\sigma(x))_{x\in\mathbb Z^d}\) be i.i.d. Suppose \(\mathbf E\sigma(0)=1\), \(\operatorname{Var}(\sigma(0))\ge\nu_0^2>0\), and \(\mathbf E e^{\theta_0|\sigma(0)-1|}\le K_0\). There are \(c>0\) and \(t_0<\infty\), depending only on \(d,\nu_0,\theta_0,K_0\), such that for every integer \(t\ge t_0\), \[\{x:u_t(x)>c h(t)\}\text{ contains an infinite nearest-neighbor component almost surely},\] where \(h(t)=t^{(4-d)/4}\) for \(d\in\{2,3\}\), \(h(t)=\log t\) for \(d=4\), and \(h(t)=(\log t)^{2/d}\) for \(d\ge5\).

Proof mechanism

The recursion is the dynamic-programming equation for a random-walk optimal stopping problem: \[2d\,u_t(x)=\sup_{\tau\le t}\mathbf E_x\sum_{k<\tau}(\sigma(X_k)-1),\] where \(X\) is a simple random walk started at \(x\) and the supremum is over stopping times of the walk. Removing the positive part gives the membrane field \(V_t\). In dimensions \(2\) and \(3\), the parabolic scaling limit of the odometer is a Brownian optimal-stopping value. Stopping the Brownian motion when it exits a ball of radius \(s\) bounds this value below by a stationary Gaussian field of finite range proportional to \(s\); RSW estimates and sprinkling give crossings at a fixed radius, and a zero-one law extracts finitely many deterministic radii. In dimension \(4\), the crossings are built directly on the lattice, for a Green field killed on exiting a ball. For \(d\ge5\), square summability of the Green function makes the odometer nearly independent across well-separated regions, and a multiscale argument rules out barriers around the origin formed by sites with \(u_t(x)<\mathbf Eu_t(0)/2\). Finally, raising every mass of a mean-\(\rho\) field by \(1-\rho\) gives a field of mean one, and \[0\le\widehat u_t^{(\rho)}(x)-u_t^{(\rho)}(x)\le(1-\rho)t/(2d)\] places the percolating level set of the mean-one field inside \(\mathcal T^{(\rho)}\) when \(\rho\) is close enough to \(1\).

A. Bou-Rabee and C. Panagiotis, “Quantitative explosion and percolation of the divisible sandpile” (2026), arXiv:2609.02829.

Rigidity of harmonic functions on the supercritical percolation cluster

Discrete unique continuation

Rigidity of harmonic functions on the supercritical percolation cluster

Definition. Fix \(d\ge2\). In Bernoulli bond percolation on \(\mathbb Z^d\), each edge is open independently with probability \(p\in(p_c(d),1)\), where \(p_c(d)\) is the bond percolation threshold; there is then almost surely a unique infinite connected component \(\mathcal C_\infty\) of open edges. Its graph Laplacian is \(\Delta_{\mathcal C_\infty}u(x)=\sum_{y\sim x}(u(y)-u(x))\), where the sum is over the open edges \(\{x,y\}\) of \(\mathcal C_\infty\), and \(B_R=\{x\in\mathcal C_\infty:|x|_2<R\}\). A function \(u\) is Lipschitz if there is \(L<\infty\) such that \(|u(x)-u(y)|\le Ld_{\mathcal C_\infty}(x,y)\) for all \(x,y\in\mathcal C_\infty\), where \(d_{\mathcal C_\infty}\) is the graph distance on the cluster.

Bond percolation. Each edge of \(\mathbb Z^2\) is open independently with probability \(p\). Here \(p_c(2)=1/2\): for \(p\le p_c(2)\) there is almost surely no infinite open cluster, whereas for \(p>p_c(2)\) there is almost surely a unique infinite open cluster. The plot shows, for one sample on a 64-by-64 grid of vertices at each of 41 values of \(p\), the fraction of the 4,096 vertices that lie in the largest open cluster.

\(p = 0.35 < p_c\). The largest open cluster has 74 of the 4,096 vertices: fraction 0.0181.

\(p = 0.50 = p_c\). The largest open cluster has 1,685 of the 4,096 vertices: fraction 0.4114.

\(p = 0.65 > p_c\). The largest open cluster has 3,964 of the 4,096 vertices: fraction 0.9678.

Largest-cluster fraction. For each displayed sample, the number of vertices in its largest open cluster divided by 4,096. The dashed line is \(p_c=1/2\).

Cluster

Pace

Run

Loading cluster

Relaxation sweeps: 0
Vertices: 0
Mean vertex displacement: 0
Maximum over interior vertices of the distance from a vertex to the average of its neighbours: —

Mean endpoint displacement / maximum vertex displacement in the harmonic embedding: 01

The largest open cluster of a bond-percolation sample on a 48-by-48 box, p = 0.62 or 0.85. Boundary vertices (yellow): the cluster vertices on the boundary of the box; they are fixed. One sweep replaces every interior vertex by the midpoint of its position and the average position of its neighbours. Edge colour: the mean displacement of its endpoints. When the residual readout falls below 10⁻⁵, the harmonic embedding, computed by a direct linear solve, is displayed.

Values

Run

Integer u: Aₙ₊₁ divides u(t) − u(s)

Gadget \(T_n\): 1
\(R_n=A_{n+1}/B_n\): 3 / 1
Lower bound for \(|u(t)-u(s)|\) when \(u\) is integer-valued and \(u(t)\ne u(s)\): 3 > 3^0

\(h\)

Tₙ: vertices (x, 1) for x = 1, …, 4 and (x, y) for x = 2, 3, y = 1, …, n, with nearest-neighbour edges and unit resistances; s = (1, 1), t = (4, 1), a = (2, 1). The effective resistance between s and t is Rₙ = Aₙ₊₁/Bₙ in lowest terms, with R₁ = 3 and Rₙ₊₁ = (3Rₙ + 2)/(Rₙ + 1). If u is harmonic at the non-terminal vertices and u(s) = 0, then u(a) = Bₙ u(t)/Aₙ₊₁. Hence an integer-valued harmonic u with u(t) ≠ u(s) satisfies |u(t) − u(s)| ≥ Aₙ₊₁ > 3ⁿ⁻¹. Vertex colour: h, the solution with h(s) = 0 and h(t) = 1, from 0 (pink) to 1 (blue). With Aₙ₊₁h selected, the labels are the integers Aₙ₊₁h.

Theorem. Let \(d\ge2\) and \(p_c(d)<p<1\), and let \(\mathcal C_\infty\) be the unique infinite cluster of Bernoulli bond percolation with parameter \(p\) on \(\mathbb Z^d\). Almost surely, if \(u:\mathcal C_\infty\to\mathbb R\) is harmonic and Lipschitz with respect to the cluster graph distance, then \(u\) is constant. (Bou-Rabee–Cooperman–Dario, arXiv:2303.04736, Theorem 1.1)

Theorem. Let \(d\ge2\) and \(p_c(d)<p<1\), and let \(\mathcal C_\infty\) be the unique infinite cluster of Bernoulli bond percolation with parameter \(p\) on \(\mathbb Z^d\). Almost surely, if \(u:\mathcal C_\infty\to\mathbb Z\) is harmonic and satisfies \(u(x)=O(|x|)\), then \(u\) is constant. (Bou-Rabee–Cooperman–Dario, arXiv:2303.04736, Theorem 1.2)

Theorem. Let \(p_c(2)<p<1\), and let \(\mathcal C_\infty\) be the unique infinite cluster of Bernoulli bond percolation with parameter \(p\) on \(\mathbb Z^2\). Almost surely, for every finitely supported \(f:\mathbb Z^2\to\mathbb Z\), if \(u:\mathcal C_\infty\to\mathbb R\) satisfies \(-\Delta_{\mathcal C_\infty}u=f|_{\mathcal C_\infty}\) and \[\limsup_{R\to\infty}R\sup_{B_R\setminus B_{R/2}}|u|=0,\] then \(u\) has finite support. (Bou-Rabee–Cooperman–Dario, arXiv:2303.04736, Theorem 1.3)

Details

Proof idea. By the large-scale regularity theory of Armstrong and Dario (2018), every harmonic function of at most linear growth on the cluster can be written uniquely as \(a+\xi\cdot x+\chi_\xi(x)\), where \(\chi_\xi\) is the corrector; \(\ell_\xi=\xi\cdot x+\chi_\xi\) is the corrected plane. For the first theorem, ergodicity reduces to the slope \(e_1\), and a modification of the environment involving a finite number of edges forces \(\ell_{e_1}\) to have an arbitrarily large Lipschitz constant. For the second theorem, the gadget below forces an integer-valued corrected plane to oscillate exponentially, which the bounds on the corrector exclude. For the third theorem, homogenization of the Green function gives the leading term \(\sum_i\big(\sum_y f(y)\ell_{e_i}(y)\big)\partial_i\bar G\) of \(u\), which decays like \(|x|^{-1}\); when its coefficients vanish, a martingale sensitivity argument and topological obstructions in the level sets of \(u\), which use planarity, show that \(u\) has finite support.

The gadget. The effective resistance of \(T_n\) satisfies \(R_1=3\) and \(R_{n+1}=(3R_n+2)/(R_n+1)\). In lowest terms \(R_n=A_{n+1}/B_n\), where \(A_0=A_1=1\), \(B_0=0\), \(B_1=1\), and \(A_m=4A_{m-1}-A_{m-2}\), \(B_m=4B_{m-1}-B_{m-2}\) for \(m\ge2\); moreover \(A_{n+1}>3^{n-1}\). Harmonicity gives \(u(a)-u(s)=B_n(u(t)-u(s))/A_{n+1}\), so if \(u\) is integer-valued and \(u(t)\ne u(s)\), then \(A_{n+1}\) divides \(u(t)-u(s)\) and \(|u(t)-u(s)|>3^{n-1}\). By finite energy, for every large \(r\), with probability at least \(c\exp(-Cr^d)\) the cluster in a cube of side \(r\) contains a copy of \(T_r\) on which \(\ell_\xi\) is not constant; an integer-valued \(\ell_\xi\) would then oscillate by at least \(c3^r\) there, which for \(r\) of order \(\log N\) contradicts the stretched-exponential bound on the probability that the corrector oscillates by \(N^2\) on a cube of side \(N\).

Each coordinate of the harmonic embedding of the infinite cluster is a corrected plane \(\ell_{e_i}\), which differs from \(x_i\) by a sublinear corrector; by the first theorem, no coordinate is Lipschitz.

If a vertex has degree one, harmonicity forces its value to equal its neighbour’s value. Consequently, a finite tree attached to the rest of the cluster at one vertex is mapped by the harmonic embedding to the image of that vertex. Benjamini, Duminil-Copin, Kozma and Yadin (2015) showed that the space of linearly growing harmonic functions on the cluster has the same dimension as that of \(\mathbb R^d\), and asked whether there are nontrivial Lipschitz harmonic functions on the cluster; the first theorem answers this question. The proofs rely on the large-scale regularity of harmonic functions on the cluster established by Armstrong and Dario (2018).

The two displayed 48-by-48 examples solve, on the largest open cluster of the sample, which meets all four sides of the box, the Dirichlet problem whose boundary values are the lattice positions of the cluster vertices on the boundary of the box. At \(p=.62\), 2,195 vertices and 2,820 edges have mean displacement 2.05 and maximum displacement 12.88; the linear-system residual is \(1.46\times10^{-14}\). At \(p=.85\), 2,302 vertices and 3,848 edges have mean displacement 0.57 and maximum displacement 2.70; the residual is \(4.14\times10^{-14}\).

A. Bou-Rabee, W. Cooperman and P. Dario, “Rigidity of harmonic functions on the supercritical percolation cluster”, Trans. Amer. Math. Soc. 378 (2025), arXiv:2303.04736. S. Armstrong and P. Dario, Comm. Pure Appl. Math. 71 (2018), arXiv:1609.09431. The quenched invariance principle on the cluster is due to N. Berger and M. Biskup, and independently P. Mathieu and A. Piatnitski, Proc. Roy. Soc. A 463 (2007).

Unique continuation on planar graphs

Discrete unique continuation

Unique continuation on planar graphs

Definition. A periodic planar graph \(\mathcal G=(\mathcal V,\mathcal E)\) is a connected graph for which there exists an embedding into the plane \(\mathbb R^2\) which is invariant under translation by a two-dimensional lattice \(L\). The weighted graph Laplacian is \(\Delta f(u)=\sum_{w\sim u}\mathbf a(u,w)(f(u)-f(w))\), where the sum is over the vertices \(w\) adjacent to \(u\) and the conductance \(\mathbf a\) is a strictly positive function on the undirected edges; \(f\) is harmonic when \(\Delta f=0\). A vertex closest to zero is designated the origin \(o\); \(B_n\) is the graph-metric ball of radius \(n\) centred at \(o\), and \(S_n\) is the set of vertices at graph distance \(n\) from \(o\).

Graph

Run

Harmonic function

Sign changes of \(f\) around \(S_{2n+1}\): 2
Edges of \(B_{2n+1}\) across which \(f\) changes sign: 0
Fraction of the vertices of \(B_{2n}\), and of \(B_n\), with no incident edge across which \(f\) changes sign: 0.0% · 0.0%

Sign of \(f\)

Unit conductances on graph-metric balls of the square, triangular, and honeycomb lattices, with \(n=19,15,22\), respectively. The boundary values \(f_k(b)=100\sin(k\theta_b)\) on \(S_{2n+1}\) are extended to the function \(f_k\) that is harmonic in \(B_{2n}\); \(\theta_b\) parametrizes \(S_{2n+1}\) by harmonic measure from \(o\). Blue and pink show the sign of \(f_k\), with zero counted as positive. Light cell sides are the edges across which \(f_k\) changes sign, dots mark the sign changes of \(f_k\) around \(S_{2n+1}\), and the dashed curve is \(S_n\). Play passes through \(k=1,\ldots,K\), with \(K=46,69,49\), by convex combinations of \(f_k\) and \(f_{k+1}\); Next increases \(k\) by one and pauses.

Theorem. Let \(\mathcal G\) be a periodic planar graph and let the positive conductances \(\mathbf a\) be invariant under its period lattice \(L\). There exists \(\varepsilon=\varepsilon(\mathbf a)>0\) such that every harmonic \(f\) satisfying \[\lim_{n\to\infty}\frac{|\{x\in B_n:|f(x)|\le1\}|}{|B_n|}\ge1-\varepsilon\] is constant. (Bou-Rabee–Cooperman–Ganguly, arXiv:2309.13728, Theorem 1.1)

Theorem. Put \(F=\mathcal G/(2L)\). If \(0<\lambda<\mathbf a(e)<\Lambda<\infty\) for every edge \(e\), there are constants \(\varepsilon_0(F,\Lambda/\lambda)>0\), \(n_0(F)\), and \(A(F,\Lambda/\lambda)>0\) such that, for \(n\ge n_0\) and \(0<\varepsilon<\varepsilon_0\), \[\Delta f=0\text{ on }B_{2n},\qquad |\{x\in B_{2n}:|f(x)|>1\}|\le\varepsilon|B_{2n}|\] imply \(\max_{B_n}|f|\le \exp(A\sqrt\varepsilon\,n)\). (Bou-Rabee–Cooperman–Ganguly, arXiv:2309.13728, Theorem 1.2)

Details

Topological lemma. Let \(\gamma\) be a cycle in \(\mathcal G\), and let \(Z,P,M\subseteq\mathcal V\) be disjoint sets such that \(Z\subseteq\gamma\); \(P\) and \(M\) are contained in the union of \(\gamma\) and the finite component of \(\mathcal G\setminus\gamma\); for every \(z\in Z\) there is a path \(\beta_z\) from \(z\) along a face of \(\mathcal G\) adjacent to \(z\), disjoint from \(P\cup M\), whose last vertex is adjacent to a vertex of \(P\) and a vertex of \(M\); and every connected component of the induced subgraphs \(\mathcal G[P]\) and \(\mathcal G[M]\) intersects \(\gamma\setminus Z\). If \(m_F\) is the size of the largest face of \(\mathcal G\), then \[|Z|\le\alpha_F|\gamma|,\qquad \alpha_F=\left(1+\frac1{2m_F^4}\right)^{-1}<1.\] The proof chooses \(Z'\subseteq Z\) with \(|Z'|\ge|Z|/m_F^4\) whose paths \(\beta_z\) are pairwise disjoint, and uses the Jordan curve theorem to find at least \(|Z'|/2\) distinct vertices of \(\gamma\setminus Z\).

Proof of the first theorem. The second theorem follows from the lemma, applied with the set where \(|f|\) is at most a suitable level \(A\) in place of the zero set of \(f\), and a covering argument. Following Buhovsky, Logunov, Malinnikova and Sodin (2017), a three-ball inequality, which the paper proves on periodic graphs, shows that a nonconstant harmonic function with \(|f|\le1\) on a fraction at least \(1-\varepsilon\) of every large square grows at least like \(\exp(bN)\) on the square of side \(2N+1\), with \(b>0\) independent of \(\varepsilon\). For \(\varepsilon\) small, this exponential lower bound and the upper bound of the second theorem give the first theorem.

A. Bou-Rabee, W. Cooperman and S. Ganguly, “Unique continuation on planar graphs”, Discrete Analysis 2025:16, arXiv:2309.13728.

Brownian motion in an incompressible random drift

Quantitative homogenization

Brownian motion in an incompressible random drift

A Brownian particle advected by a random, divergence-free velocity field.

The particle position solves \(dX_t=f(X_t)\,dt+\sqrt{2\nu}\,dW_t\), where \(\nu>0\) is the molecular diffusivity, \(W\) is a standard Brownian motion and \(f\) is a stationary random vector field with \(\nabla\cdot f=0\).

View

Run

Time 0.0

The background shows level lines of the stream function \(\psi\), shaded by the sign of \(\psi\); in Follow, arrows show the drift \(\nabla^\perp\psi\). Follow keeps the spatial scale fixed around the particle; Whole path rescales the view to contain the entire path. Replay reruns the same Brownian increments; New path draws new Brownian increments in the same field.

The finite simulation

The path is the Euler–Maruyama approximation, with step \(\Delta t=0.004\), of \(dX=\nabla^\perp\psi\,dt+\sqrt{2\nu}\,dW\) with molecular diffusivity \(\nu=0.30\). The stream function \(\psi\) is a fixed, 48-periodic random Fourier series with four modes in each of five frequency bands \(j=0,\dots,4\); the modes of band \(j\) have wavelength about \(48\cdot2^{-j}\), and every band has the same variance. Playback advances time at a constant rate. The theorems below concern stream matrices that are sums of independent pieces indexed by infinitely many triadic scales; this periodic field is not of that form.

Positive Hurst exponent

Vary the Hurst exponent \(\gamma\)

Hurst exponent \(\gamma\)

View

Run

Time 0.0

The background shows level lines of the stream function \(\psi\), shaded by the sign of \(\psi\); in Follow, arrows show the drift \(\nabla^\perp\psi\). Follow keeps the spatial scale fixed around the particle; Whole path rescales the view to contain the entire path. Replay reruns the same Brownian increments; New path draws new Brownian increments in the same field.

The finite simulation

The path is the Euler–Maruyama approximation, with step \(\Delta t=0.004\), of \(dX=\nabla^\perp\psi\,dt+\sqrt{2\nu}\,dW\) with molecular diffusivity \(\nu=0.30\). The stream function \(\psi\) is a fixed, 48-periodic random Fourier series with four modes in each of five frequency bands \(j=0,\dots,4\); the modes of band \(j\) have wavelength about \(48\cdot2^{-j}\), and the variance of band \(j\) is proportional to \(2^{-2\gamma j}\). Playback advances time at a constant rate. The theorems below concern stream matrices that are sums of independent pieces indexed by infinitely many triadic scales; this periodic field is not of that form.

Definition. Fix a nonnegative radial mollifier \(\rho\in C_c^\infty(B_1)\) with \(\int_{\mathbb R^2}\rho=1\). Let \(\Phi\) be a standard Gaussian free field on \(\mathbb R^2\), the Gaussian field with covariance kernel \(-\frac1{2\pi}\log|x-y|\) acting on mean-zero test functions, set \(\psi=\rho*\Phi\), and let \(f=\nabla^\perp\psi=(\partial_2\psi,-\partial_1\psi)\). For a molecular diffusivity \(\nu>0\), the diffusion solves \(dX_t=f(X_t)\,dt+\sqrt{2\nu}\,dW_t\) with \(X_0=0\), where \(W\) is standard two-dimensional Brownian motion. Since \(f\cdot\nabla\psi=0\), the drift is tangent to the level sets of \(\psi\).

Theorem · critically correlated drift

Quenched superdiffusive invariance principle

\(\mathbf E_0^f\) denotes the expectation for the process started at \(0\), conditioned on \(f\), and \(M_f(t)=\mathbf E_0^f|X_t|^2\).

Driving Brownian motion: \((\mathbf E|\sqrt{2\nu}W_t|^2)^{1/2}=2\sqrt{\nu t}\)
Second moment: \(M_f(t)^{1/2}\sim\sqrt2\,\pi^{-1/4}\,\sqrt t(\log t)^{1/4}\) in probability
Invariance principle: \(|\log\varepsilon^2|^{-1/4}\varepsilon X_{t/\varepsilon^2}\Rightarrow\pi^{-1/4}B_t\) for \(\mathbf P\)-a.e. realization of \(f\), where \(B\) is a standard Brownian motion

The invariance principle is quenched: it holds for \(\mathbf P\)-a.e. realization of the vector field \(f\), with convergence in law under \(\mathbf P_0^f\).

(Armstrong–Bou-Rabee–Kuusi, Inventiones Mathematicae (2026), journal, arXiv:2404.01115)

The two animations above show linearly interpolated Euler–Maruyama paths, with molecular diffusivity \(\nu=0.30\), in periodic random Fourier fields with five frequency bands.

Precise statement

Hypotheses in the critical case. Let \(d\ge2\), \(\nu\in(0,1]\), and \(\square_n=(-3^n/2,3^n/2)^d\). Let \(\mathbf j_n:\mathbb R^d\to\mathbb R_{\mathrm{skew}}^{d\times d}\), \(n\ge0\), be stationary random fields with joint law \(\mathbf P\). Define \(\mathbf k(x)-\mathbf k(y)=\sum_{n\ge0}(\mathbf j_n(x)-\mathbf j_n(y))\) and \(f=\nabla\!\cdot\mathbf k=\sum_{n\ge0}\nabla\!\cdot\mathbf j_n\). Let \(X_0=0\) and let \(X\) have generator \(\mathcal L=\nabla\!\cdot(\nu I+\mathbf k)\nabla=\nu\Delta+f\cdot\nabla\). The assumptions are: (J1) \(\mathbf j_n\) has range of dependence \(3^n\sqrt d\); (J2) collections indexed by disjoint sets of scales are independent; (J3) \(\mathbf j_n\in C_{\mathrm{loc}}^{1,1}\) almost surely and, for \(s\ge1\), \[\mathbf P\!\left[\|\mathbf j_n\|_{L^\infty(\square_n)}+\sqrt d\,3^n\|\nabla\mathbf j_n\|_{L^\infty(\square_n)}+d3^{2n}\|\nabla^2\mathbf j_n\|_{L^\infty(\square_n)}>s\right]\le e^{-s^2};\] (J4) the joint law is invariant under \(\mathbf j_n\mapsto-\mathbf j_n\) and under \(\mathbf j_n(x)\mapsto R^{\mathsf T}\mathbf j_n(Rx)R\) for every signed permutation matrix \(R\); and (J5) there are \(c_*,C_*\in(0,\infty)\) such that, for \(0\le n<m\) and \(e\in\partial B_1\), \[\left|\mathbf E\!\left[\left|\nabla\Delta^{-1}\!\left(\nabla\!\cdot\sum_{\ell=n+1}^{m}\mathbf j_\ell e\right)(0)\right|^2\right]-c_*(\log3)(m-n)\right|\le C_*.\]

Write \(M_f(t)=\mathbf E_0^f|X_t|^2\), where \(f\) is fixed and the expectation is over Brownian noise. Then \(M_f(t)/(t\sqrt{\log t})\to2dc_*^{1/2}\) in \(\mathbf P\)-probability. If \(W\) is standard \(d\)-dimensional Brownian motion, then, for \(\mathbf P\)-almost every \(f\), \[|\log\varepsilon^2|^{-1/4}\varepsilon X_{t/\varepsilon^2}\Longrightarrow\sqrt{2c_*^{1/2}}\,W_t\] in law with respect to the topology of locally uniform convergence on \(C([0,\infty);\mathbb R^d)\). In the two-dimensional special case in which \(f\) is \(\nabla^\perp\) of a mollified GFF, \(c_*=1/(4\pi)\), so the coefficient \(2dc_*^{1/2}\) equals \(2/\sqrt\pi\).

Proof idea. The proof is based on a rigorous renormalization group argument in which the coarse-grained diffusivities are analyzed inductively, scale by scale: on length scales of order \(3^m\), the generator is approximated by \(\bar s_m\Delta\), where the renormalized diffusivity \(\bar s_m\) is close to \((2c_*\log3^m)^{1/2}\). This gives sharp homogenization and large-scale regularity estimates on the infinitesimal generator, which are transferred into quantitative information on the process; since the process is Feller, convergence of generators gives convergence of the processes.

Hypotheses for positive Hurst exponent. Let \(d\ge2\), \(\gamma\in(0,1/4]\), \(\nu>0\), \(\square_0=(-1/2,1/2)^d\), and let \((\mathbf j_k)_{k\in\mathbb Z}\) be independent antisymmetric matrix fields satisfying \(\mathbf j_k\overset{\mathrm{law}}=3^{\gamma k}\mathbf j_0(3^{-k}\cdot)\). Assume: (J1) \(\mathbf j_0\) is mean zero, stationary, and has range of dependence \(\sqrt d\); (J2) \(\mathbf j_0\in C_{\mathrm{loc}}^{1,1}\) almost surely and, for \(s\ge1\), \[\mathbf P\!\left[\|\mathbf j_0\|_{L^\infty(\square_0)}+\sqrt d\,\|\nabla\mathbf j_0\|_{L^\infty(\square_0)}+d\|\nabla^2\mathbf j_0\|_{L^\infty(\square_0)}>s\right]\le e^{-s^2};\] (J3) the joint law of \((\mathbf j_k)_{k\in\mathbb Z}\) is invariant under \(\mathbf j_k\mapsto-\mathbf j_k\) and under \(\mathbf j_k(x)\mapsto R^{\mathsf T}\mathbf j_k(Rx)R\) for every signed permutation matrix \(R\); and (J4) there is \(c_*>0\) such that \(\mathbf E[|(\nabla\Delta^{-1}\nabla\!\cdot(\mathbf j_0e))(0)|^2]=c_*\log3\) for every \(e\in\partial B_1\). Let \(\mathbf k=\sum_{k\in\mathbb Z}\mathbf j_k\) modulo constant antisymmetric matrices, and let \(X\) have generator \(\nabla\!\cdot(\nu I+\mathbf k)\nabla\).

Theorem (quenched power-law superdiffusivity). Define \[R(t)=\left((\nu t)^{2-\gamma}+c_*\gamma^{-1}t^2\right)^{1/[2(2-\gamma)]}.\] For fixed \(\mathbf k\), put \(M_{\mathbf k}(t)=\mathbf E^{\mathbf k,0}|X_t|^2\). There are \(\gamma_0=\gamma_0(d,c_*)>0\) and \(C=C(d,c_*)<\infty\) such that, if \(0<\gamma\le\gamma_0\), then, for every \(t>0\) and \(1\le p\le C^{-1}\gamma^{-1}|\log\gamma|^{-6}\), \[\mathbf E\!\left[|M_{\mathbf k}(t)-2dR(t)^2|^p\right]^{1/p}\le C\bigl(p^{1/2}+|\log\gamma|^{1/2}\bigr)\gamma^{1/2}|\log\gamma|^4R(t)^2,\] \[\mathbf E\!\left[|\mathbf E^{\mathbf k,0}X_t|^{2p}\right]^{1/p}\le C\bigl(p+|\log\gamma|\bigr)\gamma|\log\gamma|^7R(t)^2.\] The outer expectation is over \(\mathbf k\). With \(t_*=(c_*^{-1}\gamma\nu^{2-\gamma})^{1/\gamma}\), \[R(t)\sim(c_*/\gamma)^{1/[2(2-\gamma)]}t^{1/(2-\gamma)}\qquad(t/t_*\to\infty).\]

S. Armstrong, A. Bou-Rabee and T. Kuusi, “Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift”, Inventiones Mathematicae (2026), journal, arXiv:2404.01115; “Superdiffusion and anomalous regularization in self-similar random incompressible flows” (2026), arXiv:2601.22142. For \(\nabla^\perp\) of a mollified GFF in two dimensions, G. Cannizzaro, L. Haunschmid-Sibitz and F. Toninelli obtained the growth rate \(t(\log t)^{1/2}\) of the annealed second moment up to double logarithmic corrections, arXiv:2106.06264; the double logarithmic corrections were removed by G. Chatzigeorgiou, P. Morfe, F. Otto and L. Wang, “The Gaussian free field as a stream function”, Ann. Probab. 53 (2025), arXiv:2212.14244.

Brownian motion in a log-correlated potential

Quantitative homogenization

Brownian motion in a log-correlated potential

A Brownian particle in \(\mathbb R^d\), \(d\ge2\), whose drift is the gradient of a log-correlated random potential.

The potential has independent fluctuations of equal strength at every length scale. The drift pushes the particle toward the large values of the potential, where the reversible measure puts its mass, and to travel the particle must cross the barriers of the energy landscape given by minus the potential. As more scales contribute, their effects accumulate and the effective diffusivity tends to zero. The particle is then subdiffusive: the time it needs to travel a large distance \(r\) grows faster than \(r^2\).

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Amplitude \(\sigma\)

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Time: 0.0
Distance from start: 0.0
Root-mean-square distance of \(\sqrt2\,W_t\): 0.0

One numerical path · fixed scale

One numerical path in one sample of the potential in the plane. The background colour shows the sign and size of the potential and the thin lines are its level lines; the drift is perpendicular to them and points toward larger values of the potential. The last readout is the root-mean-square distance \(\sqrt{4t}\) of \(\sqrt2\,W_t\) in the plane, the motion without drift, for comparison with this one path. Replay uses the same noise; New field samples another potential and another noise.

The finite simulation

The numerical path approximates the solution in the plane of \(dX=\nabla P(X)\,dt+\sqrt2\,dW\) by the Euler–Maruyama scheme with step \(\Delta t=0.004\), starting at the origin. The potential is \(P=\sigma\sum_{k=1}^{4}g_k\), where the \(g_k\) are independent smooth random fields and \(g_k\) has the law of \(g_1(3^{1-k}\,\cdot)\); the three buttons set \(\sigma=1.2,\,2.4,\,3.6\). Playback advances the time \(t\) at a constant rate. Four scales illustrate the model; the theorem below concerns infinitely many.

Definition. Let \(d\ge2\) and let \(\mathsf g_0,\mathsf g_1,\mathsf g_2,\ldots\) be independent random fields on \(\mathbb R^d\) such that \(\mathsf g_k\) has the law of \(\mathsf g_0(3^{-k}\,\cdot)\). The field \(\mathsf g_0\) is stationary, has mean zero and range of dependence \(\sqrt d\), is \(C^{1,1}_{\mathrm{loc}}\), and its size is controlled by the disorder strength \(\delta\in(0,1/2]\); the precise assumptions are (g1)–(g4) of the paper. Set \[\mathbf a(x)=\exp\Big(\sum_{k\ge0}\big(\mathsf g_k(x)-\mathsf g_k(0)\big)\Big).\] The particle solves \[dX_t=\nabla\log\mathbf a(X_t)\,dt+\sqrt2\,dW_t,\] where \(W\) is a standard Brownian motion. The diffusion has generator \(\mathbf a^{-1}\nabla\cdot(\mathbf a\nabla)\) and reversible measure \(\mathbf a(x)\,dx\), and \(\log\mathbf a\) is log-correlated: the variance of \(\log\mathbf a(x)-\log\mathbf a(y)\) lies between two positive multiples of \(\log|x-y|\) for all sufficiently large \(|x-y|\). The environment is a realization of the fields; the law of the trajectories in a fixed environment is called quenched, and its average over the environment annealed.

Theorem. Assume (g1)–(g4). There is \(\delta_0(d)>0\) such that, if \(0<\delta\le\delta_0\), the following holds. Let \((\mathsf g_j)_{j\in\mathbb Z}\) be independent, with \(\mathsf g_j\) having the law of \(\mathsf g_0(3^{-j}\,\cdot)\). There is a family of quenched path laws, measurably determined by \((\mathsf g_j)_{j\in\mathbb Z}\), which almost surely defines a continuous strong Markov process \(Z\) with the following properties. For every \(x\), as the integer \(N\to\infty\), the annealed law of \(\big(3^{-N}X_{3^{2N}t/\bar{\mathbf a}_N}\big)_{t\ge0}\) with \(X_0=3^Nx\) converges to the annealed law of \(Z\) started at \(x\), where \(\bar{\mathbf a}_N\) is the effective diffusivity at scale \(3^N\). Almost surely, \(Z\) is reversible with respect to a measure that is singular with respect to Lebesgue measure. There are \(C<\infty\) and \(\eta>0\), depending only on \(d\) and the law of \(\mathsf g_0\), such that, for every starting point and every \(r\in(0,1]\), the annealed mean time for \(Z\) to leave the cube of side \(r\) centered at its starting point is at most \(Cr^{2+\eta}\); in dimension two one can take \(\eta=\tau^2/\log3\), where \(\tau^2=\log\mathbf E[e^{\mathsf g_0(0)}]\). (Armstrong–Bou-Rabee–Kuusi, Theorem A)

Consequently, for every \(x\), the annealed path law of \(Z\) started at \(x\), and almost surely its quenched path law, is mutually singular with every mixture of Brownian laws with arbitrary covariance matrices. (Theorem 7.1(iii))

The effective diffusivity

Keeping only the fields \(\mathsf g_0,\ldots,\mathsf g_L\) gives a diffusion \(X^{(L)}\), and its effective diffusivity \(\bar{\mathbf a}_L\) is characterized by the annealed scaling limit \(\bigl(\varepsilon X^{(L)}_{t/\varepsilon^2}\bigr)_{t\ge0}\to\bigl(\sqrt{2\bar{\mathbf a}_L}\,W_t\bigr)_{t\ge0}\) in law as \(\varepsilon\downarrow0\), with \(X^{(L)}\) started at the origin. These diffusivities are nonincreasing in \(L\), and \(e^{-(L+1)\tau^2}\le\bar{\mathbf a}_L\le3^{-\eta_{\mathrm{sub}}(L+1)}\) for every \(L\ge0\), for some \(\eta_{\mathrm{sub}}>0\) depending only on \(d\) and the law of \(\mathsf g_0\); in dimension two \(\bar{\mathbf a}_L=e^{-(L+1)\tau^2}\) exactly. When \(\mathsf g_0\) is Gaussian, the reversible measure of \(Z\) is Gaussian multiplicative chaos. In the incompressible models of Brownian motion in an incompressible random drift, the effective diffusivity increases with scale; here it decreases.

S. Armstrong, A. Bou-Rabee, and T. Kuusi, “Anomalous scaling limit of a Brownian particle in a log-correlated potential” (2026), arXiv:2610.12335 · Lean formalization.

Quantitative Einstein relation

Quantitative homogenization

Quantitative Einstein relation

A particle diffusing in a disordered medium, subjected to a small constant force, eventually drifts at an asymptotic velocity.

Time runs from left to right. Each gold path is driven by the same Brownian motion as its blue partner, with an added constant force.

No forceSmall constant force32 pairs of paths

Time0

Force \(\lambda\)

Run

The vertical coordinate is the first coordinate of the position, in the direction \(e_1\) of the force. Thin lines are individual paths; the thick blue and gold lines are the two sample means. The vertical scale is fixed for the whole run. Drag the time bar to choose a time; Replay reruns the same Brownian increments.

The Einstein relation and this simulation

The Einstein relation states that, as the force tends to zero, the ratio of the limiting velocity to the force magnitude converges to the diffusivity matrix of the unforced particle, evaluated in the force direction. Here \(\widehat\Sigma_{11,t}\) is the unbiased sample variance of the first coordinate of the 32 unforced paths at time \(t\), divided by \(t\), and \(\widehat\mu_t\) is the difference of the two sample means of the first coordinate, divided by \(\lambda t\).

Diffusivity \(\widehat\Sigma_{11,t}\): —
Mobility \(\widehat\mu_t\): —

We integrate 32 pairs from the origin with \(dX=-\nabla V(X)\,dt+dB\), \(dY=(-\nabla V(Y)+\lambda e_1)\,dt+dB\), Euler–Maruyama step 0.025, and \(V(x,y)=0.46\sin(x+0.23)+0.31\cos(y-0.41)+0.22\sin(x+0.73y)+0.15\cos(1.7x-0.8y)\). Every run lasts until time \(4\lambda^{-2}\); playback speed changes with the force. The potential is a fixed deterministic function, not a stationary random field with finite range of dependence as in the theorem. At finite \(\lambda\) and \(t\) the two estimates need not be equal, and neither need be monotone in \(t\).

Definition. On a probability space \((\Omega,\mathcal F,\mathbb Q)\), let \(d\ge2\), \(\Lambda\ge1\), let \(r_0\ge1\) be an integer, let \(\mathbf a\) be a symmetric positive-definite matrix field, and let \(V\) be a scalar field. Assume \((\mathbf a,V)\) is stationary with finite range of dependence \(r_0\), \(\Lambda^{-1}|\xi|^2\le\xi\cdot\mathbf a(x)\xi\le\Lambda|\xi|^2\), \(|V(x)|\le\Lambda\), and both fields are \(\Lambda\)-Lipschitz. The unforced generator is \(\mathcal Lf=\tfrac12e^{2V}\nabla\!\cdot(e^{-2V}\mathbf a\nabla f)\). For \(\mathbb Q\)-almost every environment, \(\varepsilon X_{t/\varepsilon^2}\Rightarrow W_t\) in law, where \(\mathbf E[W_t\otimes W_t]=t\Sigma_X\) for a deterministic effective diffusivity \(\Sigma_X\). Applying a constant force of magnitude \(\lambda\in(0,1]\) in the direction \(e_1\) gives \(\mathcal L^\lambda f=\mathcal Lf+\lambda\mathbf a e_1\!\cdot\nabla f\). The resulting biased diffusion \(X^\lambda\), started at the origin, satisfies a quenched law of large numbers: for almost every environment, \(X_t^\lambda/t\to\ell_X(\lambda)\) almost surely, for a deterministic effective velocity \(\ell_X(\lambda)\).

Theorem. Under the hypotheses above, there is a deterministic \(\beta^*=\beta^*(d)\in(0,1/4)\) and a random scale \(\lambda_0:\Omega\to(0,1]\) such that, \(\mathbb Q\)-almost surely, \[\left|\frac{\ell_X(\lambda)}{\lambda}-\Sigma_Xe_1\right|\le\lambda^{\beta^*}\qquad(0<\lambda\le\lambda_0).\] There are \(C,c>0\) and \(\sigma\in(0,1)\), depending only on \((d,\Lambda,r_0)\), such that \(\mathbb Q(\lambda_0\le\lambda)\le C\exp[-c(\log(1/\lambda))^\sigma]\) for every \(\lambda\in(0,1]\). (Bou-Rabee–Xu, arXiv:2605.26082, Theorem 2)

Question. Prove the conjectured sharp rate \(\lambda\) for \(d\ge3\) and \(\lambda\sqrt{\log(1/\lambda)}\) for \(d=2\).

Details

Proof idea. Under a force of magnitude \(\lambda\), the ballistic displacement \(\lambda t\) and the diffusive fluctuation \(\sqrt t\) are comparable precisely when \(t\asymp\lambda^{-2}\), that is, on spatial scales of order \(\lambda^{-1}\). A time change absorbs the potential into the divergence-form coefficient \(e^{-2V}\mathbf a\). On a cube of side slightly larger than \(\lambda^{-1}\), the solution of a resolvent equation, via the Feynman–Kac formula, encodes the mobility error as an exponentially weighted average of the path.

This average splits into four contributions: short-time, long-time, exit, and homogenization. The short-time and exit terms are controlled directly; the long-time term by a law of large numbers uniform in \(\lambda\), and the homogenization term by a quantitative estimate for the tilted resolvent. Optimizing the free parameters in the resulting bound yields the algebraic rate \(\lambda^{\beta_*}\).

A. Bou-Rabee and R. Xu, “Quantitative Einstein relation for reversible diffusions in a random environment” (2026), arXiv:2605.26082.

Random walk on orthogonal tilings

Quantitative homogenization

Random walk on orthogonal tilings

Definition. Cover \(D\subseteq\mathbb R^d\) by closed, convex polytopes \(P_v\) with disjoint interiors, each associated with a vertex \(v\) in its interior, such that each compact subset of \(D\) intersects finitely many polytopes. Vertices \(v\) and \(w\) are joined by an edge, \(v\sim w\), if \(P_v\cap P_w\) is a \((d-1)\)-dimensional facet. The tiling is orthogonal if, for each edge, \(w-v\) is orthogonal to the hyperplane containing \(P_v\cap P_w\). Put \(a(v,w)=\operatorname{vol}_{d-1}(P_v\cap P_w)/\lVert v-w\rVert\), \(a(v)=\sum_{w\sim v}a(v,w)\), and \(p(v,w)=a(v,w)/a(v)\).

Pace

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Loading tessellation

Steps: 0
Visited cells: 0
Area of visited cells / total area: —
Unvisited cells: 3,000

First-visit order: firstlast

The trace of random walk on the Voronoi tessellation of 3,000 points in a square. The points lie in 3,000 distinct cells of a 96-by-96 grid, drawn without replacement with probabilities proportional to the mass that a three-dimensional Gaussian multiplicative chaos measure (γ = 1.4) on a cube gives to the part of a central slab, of thickness 1/24 of the cube, above each grid cell. The point set is reflected across the four sides of the square before the tessellation is computed, so every cell is bounded; a step toward a reflected point leaves the walk in place. From cell v the walk moves to an adjacent cell w with probability proportional to a(v, w) = length(P_v ∩ P_w)/|v − w|. Colour: first-visit order on a logarithmic scale. In Watch with Follow, cyan spokes show the next-step probabilities and the current cell is outlined in gold.

Theorem. Let \(D\subseteq\mathbb R^d\), let \(U\subset\mathbb R^d\) be a bounded Lipschitz domain with \(\overline U\subset D\), and let \(G_n=(V_n,E_n)\) be orthogonal tilings of \(D\). For \(z\in D\), let \(z^n\) be the nearest vertex of \(V_n\) to \(z\), with ties broken in lexicographical ordering, and put \(\varepsilon_n=\sup_{v\in V_n}\operatorname{diam}(P_v)\). Assume \[\varepsilon_n+\sup_{z\in D}\lVert z-z^n\rVert\longrightarrow0.\] Assume also at least one of: (I) \(d=2\); (II) \(\inf_{v\in V_n}\operatorname{vol}_d(P_v)\ge\exp(-o(\varepsilon_n^{-1}))\); or (III) for some \(C,\alpha>0\), uniformly in \(n\) and \(v\), \[\operatorname{diam}(P_v)\le C\max_{w\sim v}\!\left(\lVert w-v\rVert\operatorname{vol}_{d-1}(P_v\cap P_w)\right)^\alpha.\] Give the walk transition probabilities \(p(v,w)=a(v,w)/\sum_{y\sim v}a(v,y)\), where \(a(v,w)=\operatorname{vol}_{d-1}(P_v\cap P_w)/\lVert v-w\rVert\). For each \(z\in U\), its linearly interpolated path, started at \(z^n\) and stopped upon exiting \(U\), converges in law to standard Brownian motion started at \(z\) and stopped upon exiting \(U\), with respect to the metric on curves viewed modulo time parametrization, \[d(\phi_1,\phi_2)=\inf_\psi\sup_{t\in[0,T_1]}|\phi_1(t)-\phi_2(\psi(t))|\] for continuous curves \(\phi_i:[0,T_i]\to\mathbb R^d\), where the infimum is taken over all increasing homeomorphisms \(\psi:[0,T_1]\to[0,T_2]\). The convergence is uniform over all choices of \(z\). (Bou-Rabee–Gwynne, arXiv:2405.11673, Theorem A)

Theorem. For all \(d\ge3\), there exists a sphere packing of \(\mathbb R^d\) upon which simple random walk with unit conductance does not converge to Brownian motion modulo time parametrization in the scaling limit. Moreover, the spheres all have radii either \(1\) or \(1/2\) and are tangent to a bounded number of other spheres. (Bou-Rabee–Gwynne, arXiv:2405.11673, Theorem 6.1)

Details

Proof idea. The main contribution is an \(L^2\) to \(L^\infty\) estimate for the operator \(\sum_{w\sim v}a(v,w)(h(w)-h(v))\). By an \(L^2\) bound and a Poincaré-type inequality, the discrete harmonic function \(h_D\) is pointwise close to the continuum one \(h_C\) on a dense set of columns; repeating this on the remaining bad set, using (II) or (III) to close the loop, yields the pointwise bound. When \(d=2\), closeness on a dense set of columns and the maximum principle give the pointwise bound everywhere. Uniform convergence of the Dirichlet problem implies convergence of harmonic measure, and the strong Markov property of the walk with an approximation argument gives convergence of the trace.

In two dimensions the argument gives a new, short proof of the main result of Gurel-Gurevich, Jerison and Nachmias (2020), the convergence of the Dirichlet problem on orthodiagonal maps, without requiring the domains to be simply connected.

A. Bou-Rabee and E. Gwynne, “Random walk on sphere packings and Delaunay triangulations in arbitrary dimension”, Proc. Lond. Math. Soc. 131 (2025), arXiv:2405.11673. O. Gurel-Gurevich, D. Jerison and A. Nachmias, “The Dirichlet problem for orthodiagonal maps”, Adv. Math. 374 (2020).

Internal DLA on cylinders

Internal DLA and divisible sandpile

Internal DLA on cylinders

Definition. Let \(V_N\) be a finite connected vertex-transitive graph and let \(P_N\) be its lazy simple-random-walk kernel. Initially occupy \(V_N\times\mathbb Z_{\le0}\). Each particle performs an independent cylinder walk started at a site chosen uniformly from \(V_N\times\{0\}\); from \((x,y)\) it moves to \((x,y+1)\) or \((x,y-1)\), each with probability \(1/4\), and otherwise moves to \((x^{\prime},y)\) with probability \(P_N(x,x^{\prime})/2\). It settles at the first unoccupied vertex.

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Particles: 0
Height \(n/N\): 0
Maximal fluctuation: 0
\(N^{-1}\sum_{\mathrm{settled}}\sqrt2\cos(2\pi x/N)\): 0.00

Growth: settlement order · Field: coral = occupied above \(n/N\), cyan = unoccupied at or below \(n/N\)

Internal DLA on Z₄₈ × Z with the kernel of the definition: from (x, y), up or down with probability 1/4 each; otherwise x moves by the lazy simple random walk on Z₄₈. 1,728 particles start uniformly on height 0; when a walk steps below height 0, its position on returning to height 0 is sampled from the exact law of that position. Colour: settlement order. Dashed line: height n/48 after n particles. Field: coral, occupied sites above n/48; cyan, unoccupied sites at or below n/48. Readouts: n/48; max{h_out − n/48, n/48 − h_in}, where h_out is the highest occupied height and h_in the highest fully occupied height; and (1/48) Σ √2 cos(2πx/48) over occupied sites (x, y) with y > 0.

Base

Run

Particles: 0
Height \(n/N\): 0.0
Maximal fluctuation: 0.0
Highest occupied layer: 0

Settlement order

The same process over the three bases in Figure 1 of the paper: the 15-by-15 square torus (225 vertices, degree 4), the 9-by-9 triangular torus (81 vertices, degree 6) and the Nauru graph (24 vertices, degree 3), each with its lazy simple random walk: P(x, x) = 1/2 and P(x, y) = 1/(2 deg x) for neighbours x, y. Occupied vertices above height 0 are drawn as prisms over the cells of the displayed embedding. Colour: settlement order. Readout: max{h_out − n/N, n/N − h_in}. Drag horizontally to rotate.

Common setup

For each \(N\ge2\), let \(V_N\) be a finite connected vertex-transitive graph with \(|V_N|=N\), and let \(P_N\) be its lazy simple-random-walk kernel. On \(\mathbb G_N=V_N\times\mathbb Z\), put \(R_h=V_N\times\{y\in\mathbb Z:y\le h\}\). Start IDLA from \(A(0)=R_0\), and write \(A(T)\) for the cluster after \(T\) particles.

Theorem 1.1 · maximal fluctuations

Maximal fluctuations from the rectangular shape

Parameters. Fix \(\nu>0\) and \(m\ge1\). Let \(\tau_{\mathrm{mix}}\) be the total-variation mixing time of \(P_N\) at threshold \(1/4\), and consider integers \(1\le T\le N^m\).

Scales. Write \(\log_+s=\max\{\log s,0\}\). There exists a constant \(C=C(\nu,m)<\infty\) such that the conclusion below holds with \[T^\#=N\sqrt{\tau_{\mathrm{mix}}}(\log N)^2,\] \[B_N(T)=\frac{T\wedge T^\#}{N}\left(\log N+\log_+(T/N)\right),\] \[\Delta_N(T)=C\max\!\left\{\log N,\sqrt{B_N(T)}\right\}.\] Finally put \(\ell_N(T)=\lfloor T/N-\Delta_N(T)\rfloor\) and \(u_N(T)=T/N+\Delta_N(T)\).

Conclusion

\[\mathbb P\!\left(R_{\ell_N(T)}\subset A(T)\subset R_{u_N(T)}\right) \ge1-N^{-\nu}.\]

The estimate holds uniformly over \(1\le T\le N^m\), for every sufficiently large \(N\).

(Bou-Rabee–Silvestri–Yadin, arXiv:2604.21142, Theorem 1.1)

Theorem 1.9 · Gaussian fluctuations

Scalar central limit theorem for finite-mode test functions

Eigenmodes. Let \(P_Nf_k^N=\lambda_k(N)f_k^N\), where \(1=\lambda_1(N)>\lambda_2(N)\ge\cdots\ge\lambda_N(N)\ge0\), normalized by \(N^{-1}\sum_x f_j^N(x)f_k^N(x)=\delta_{jk}\).

Discrepancy. Set \(D_T=\mathbf1_{A(T)}-\mathbf1_{R_{T/N}}\) and \(\langle\!\langle D,\phi\rangle\!\rangle=\sum_{(x,y)\in\mathbb G_N}D(x,y)\phi(x,y)\).

Test function and scaling. Fix \(y_0>0\) and an integer \(K\ge2\) independent of \(N\). For \(2\le k\le K\), let \(\alpha_k\), independent of \(N\), be differentiable with bounded derivative near \(y_0\). Let \(a_N>0\) satisfy, for every \(2\le k\le K\), \[\frac{a_N}{\log N+\log_+a_N}\longrightarrow\infty,\] \[a_N\sqrt{2(1-\lambda_k(N))}\longrightarrow\gamma_k\in(0,\infty).\] Put \(T=N\lfloor y_0a_N\rfloor\), define \(\phi_N(x,r)=\sum_{k=2}^K\alpha_k(r)f_k^N(x)\), and let \(\varphi_N(x,y)=\phi_N(x,y/a_N)\). Define \[\sigma^2(\phi)=\sum_{k=2}^K\alpha_k(y_0)^2 \frac{1-e^{-2\gamma_k y_0}}{2\gamma_k}.\]

Conclusion

\[\frac{\langle\!\langle D_T,\varphi_N\rangle\!\rangle}{\sqrt{Na_N}} \Rightarrow\mathcal N\!\left(0,\sigma^2(\phi)\right).\]

When \(V_N=\mathbb Z_n^d\), \(N=n^d\), this variance matches the height-\(y_0\) slice of the Dirichlet GFF on \(\mathbb T^d\times\mathbb R_+\), up to the constant rescaling induced by the lazy kernel. The same mode-by-mode identification applies to bases converging spectrally to a compact Riemannian manifold.

(Bou-Rabee–Silvestri–Yadin, arXiv:2604.21142, Theorem 1.9)

Details

Proof idea. When \(\varphi\) is discrete harmonic and has zero horizontal average, \(\sum_{(x,y)\in A(t)}\varphi(x,y)\) is a martingale in the particle index \(t\), and a martingale central limit theorem gives a Gaussian limit. The test function is replaced by the discrete harmonic extension of its values at the cluster's typical height; an Efron–Stein variance inequality, using the abelian property of IDLA, controls the replacement error, and the maximal fluctuations bound of Theorem 1.1 gives concentration of the cluster around its typical height.

In the displayed run the base graph is the cycle \(\mathbb Z_{48}\), and the fourth readout is \(N^{-1}\langle\!\langle D_n,f_2^N\rangle\!\rangle\) with \(f_2^N(x)=\sqrt2\cos(2\pi x/N)\).

A. Bou-Rabee, V. Silvestri and A. Yadin, “Gaussian fluctuations for Internal DLA on cylinders” (2026), arXiv:2604.21142.

Critical long-range random conductance walk

Quantitative homogenization

Critical long-range random conductance walk

Definition. Fix \(d\ge1\) and \(\lambda\in(0,1]\). Consider a random walk on \(\mathbb Z^d\) whose jump rate from \(x\) to \(y\) is \(a(x,y)|x-y|^{-(d+2)}\), where the conductances \(\{a(x,y)\}\), indexed by unordered edges \(\{x,y\}\), are i.i.d. and satisfy \(\lambda\le a(x,y)\le\lambda^{-1}\).

Pace

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Jump length at most 48

Jump-chain steps \(n\): 0
Current jump length: 0
Maximum jump length: 0
\(|X_n|/\sqrt{\max(1,n)}\): 0.00

Order of the last 2,600 jumps

Jump chain of the critical long-range random conductance model on Z², with jumps of length at most 48. From x, a displacement z with 0 < |z| ≤ 48 is proposed with probability proportional to |z|⁻⁴ and accepted with probability a(x, x+z)/1.35; a rejected proposal is drawn again. The conductances a(x, y) = a(y, x) are i.i.d. uniform on [0.65, 1.35] and fixed. Only the last 2,600 jumps are drawn, coloured by their order. Scale centres the view at the origin, with width of order √n, enlarged if needed to contain the drawn jumps; Follow frames the last 300 jumps. The truncated jumps have finite variance, so the displacement readout is scaled by √n. The theorem below concerns the untruncated continuous-time walk.

Theorem. Let \(d\ge1\), \(\lambda\in(0,1]\), and let \(\{a(x,y)\}_{x\ne y}\) be i.i.d. symmetric random conductances indexed by unordered edges, with \(a(x,y)=a(y,x)\) and \(\lambda\le a(x,y)\le\lambda^{-1}\). Define \[\mathcal L^\varepsilon h(x)=\frac{\varepsilon^d}{\kappa_\varepsilon}\sum_{z\in\varepsilon\mathbb Z^d\setminus\{0\}}a(x/\varepsilon,(x+z)/\varepsilon)\frac{h(x+z)-h(x)}{|z|^{d+2}},\qquad \kappa_\varepsilon=\varepsilon^d\!\sum_{\substack{z\in\varepsilon\mathbb Z^d\setminus\{0\}\\|z|\le1}}|z|^{-d}.\] If \(X\) is the lattice walk with jump rates \(a(x,y)|x-y|^{-(d+2)}\), then \(X_t^\varepsilon:=\varepsilon X_{t/(\varepsilon^2\kappa_\varepsilon)}\) has generator \(\mathcal L^\varepsilon\). Let \(g\in L^2(\mathbb R^d)\) be a nonnegative compactly supported probability density, put \(g_\varepsilon(x)=\varepsilon^{-d}\int_{x+[-\varepsilon/2,\varepsilon/2)^d}g(y)\,dy\), and give \(X^\varepsilon\) the initial law \(\mathbb P(X_0^\varepsilon=x)=\varepsilon^d g_\varepsilon(x)\). Let \(W\) be standard \(d\)-dimensional Brownian motion and let \(X_0\) have law \(g(x)\,dx\), independently of \(W\). Then, for \(\mathbb P\)-almost every realization of the conductances, \(X^\varepsilon\Rightarrow X_0+\sqrt{\mathbf E[a(0,e_1)]/d}\,W\) in \(D([0,\infty);\mathbb R^d)\) with the Skorokhod topology. (Bou-Rabee–Dario, arXiv:2604.21143, Theorem 1.2)

Theorem. Under the same conductance assumptions, let \(\sigma\in(0,1)\), \(\mu\ge0\), let \(U\subset\mathbb R^d\) be a bounded domain with \(C^{2,\sigma}\) boundary, and let \(f\in C^{0,\sigma}(\overline U)\). For \(\varepsilon\in(0,1)\), define \(u_\mu^\varepsilon:\varepsilon\mathbb Z^d\to\mathbb R\) and \(\bar u_\mu\in H_0^1(U)\) by \[\mu u_\mu^\varepsilon-\mathcal L^\varepsilon u_\mu^\varepsilon=f\ \text{ in }U\cap\varepsilon\mathbb Z^d,\quad u_\mu^\varepsilon=0\ \text{ on }\varepsilon\mathbb Z^d\setminus U,\] \[\mu\bar u_\mu-\frac{\mathbf E[a(0,e_1)]}{2d}\Delta\bar u_\mu=f\ \text{ in }U,\quad \bar u_\mu=0\ \text{ on }\partial U.\] There are constants \(C,c>0\), depending only on \(\mu,d,\lambda,\sigma,U\), and \(\|f\|_{C^{0,\sigma}}\), and a random variable \(\mathcal X\) such that \(\mathbb P(\mathcal X>s)\le Ce^{-cs}\) for every \(s\ge1\), and \[\|u_\mu^\varepsilon-\bar u_\mu\|_{L^2(U\cap\varepsilon\mathbb Z^d)}\le\frac{C}{\sqrt{|\log\varepsilon|}}\quad\text{whenever }|\log\varepsilon|\ge\mathcal X,\] where \(\|v\|_{L^2(U\cap\varepsilon\mathbb Z^d)}^2=\varepsilon^d\sum_{x\in U\cap\varepsilon\mathbb Z^d}|v(x)|^2\). (Bou-Rabee–Dario, arXiv:2604.21143, Theorem 1.1)

Details

Proof idea. The proof of the homogenization theorem is a two-scale expansion, following the quantitative homogenization program of Armstrong, Kuusi and Mourrat adapted to the nonlocal critical setting. The corrector estimate constructs a divergence-free comparison flux by routing long-range excess currents along canonical nearest-neighbour paths. A Poincaré inequality for the critical kernel holds with a constant independent of \(\varepsilon\), because each triadic shell of the kernel contributes the same amount to the energy. The smooth Taylor remainder is estimated in \(L^2\) by Hoeffding’s concentration inequality. The invariance principle follows by tightness of the rescaled processes, via off-diagonal heat-kernel bounds and Aldous’ criterion, and identification of the limit through an Euler approximation of the semigroup.

Displayed jump chain. For one realization of the conductances on \(\mathbb Z^2\), the jump chain uses displacements \(0<|z|\le48\) and the jump index \(n\). The generator \(\mathcal L^\varepsilon\) in the theorem sums over every nonzero displacement and is normalized by \(\kappa_\varepsilon\asymp|\ln\varepsilon|\), which normalizes the logarithmically divergent second moment.

A. Bou-Rabee and P. Dario, “Quantitative homogenization for the critical long-range random conductance model” (2026), arXiv:2604.21143.

Eulerian walkers on Z²

Random walks with memory

Eulerian walkers on Z²

Definition. At each \(v\in\mathbb Z^2\), choose \(\rho_0(v)\) independently and uniformly from the four outgoing edges. The clockwise rotor mechanism has cyclic order \(N,E,S,W\). Starting from \(X_0=0\), define \(\rho_{t+1}(X_t)\) by advancing \(\rho_t(X_t)\) once in this order, leave every other rotor unchanged, and let \(X_{t+1}\) be the head of \(\rho_{t+1}(X_t)\). Once \(\rho_0\) is fixed, the walk is deterministic. Its range is \(R_t=\{X_0,\ldots,X_t\}\).

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Steps: 0
Visited sites \(|R_t|\): 1
Circuits completed: 0
Maximum Euclidean distance of a visited site from the origin: 0
Least-squares slope of \(\log|R_s|\) against \(\log s\), for geometrically spaced \(s\in[t/10,t]\): —

First-visit circuit

Eulerian walker on Z² with the clockwise rotor mechanism and independent uniform initial rotors, started at the origin of an 801-by-801 box; the run stops if the walker reaches the boundary of the box. A circuit consists of four successive returns to the origin; the circuit readout is the number of completed circuits plus one quarter for each further return. A site is coloured by the circuit during which it was first visited. Between the completion times T(n) and T(n+1), every directed edge is used at most once, and if T(n+1) < ∞ the walk departs exactly four times from every site of the range at time T(n).

Theorem 1.1 · setup

A rotor mechanism specifies a cyclic permutation of the directed edges out of each vertex. A doubly periodic graph in \(\mathbb R^2\) is a connected, locally finite graph with vertex set \(V\subset\mathbb R^2\) on which a rank-two lattice \(\Lambda\subset\mathbb R^2\) acts by translation, the translations being automorphisms with finitely many orbits in \(V\); a rotor mechanism on it is doubly periodic if its cyclic orders are invariant under \(\Lambda\). Let \(G\) be either the square lattice with its clockwise rotor mechanism, or a doubly periodic graph in \(\mathbb R^2\) of maximum degree \(3\) with a doubly periodic rotor mechanism. At every vertex choose the initial rotor independently and uniformly among its outgoing edges, and start the walk at a fixed vertex \(o\).

Definitions

Range, circuit range, and completion time

Range. \(R_t=\{X_0,\ldots,X_t\}\).

Circuit. One circuit consists of \(\deg(o)\) successive returns to \(o\). Put \[N_o(t)=\#\{0\le s<t:X_s=o\},\] \[T(n)=\inf\{t\ge0:X_t=o\ \text{and}\ N_o(t)\ge n\deg(o)\},\] and \(A_n=R_{T(n)}\) whenever \(T(n)<\infty\), with \(A_0=\{o\}\).

Almost-sure conclusions

Recurrence. Every vertex is visited infinitely often.
Shape. There are a deterministic compact convex set \(B\subset\mathbb R^2\), containing \(0\) in its interior, and a deterministic \(\kappa>0\) such that, in Hausdorff distance, \[n^{-1}A_n\longrightarrow B,\qquad t^{-1/3}R_t\longrightarrow\kappa B.\]
Range exponent. The limit \[\lim_{t\to\infty}|R_t|t^{-2/3}\] exists, is deterministic, and belongs to \((0,\infty)\).

(Bou-Rabee–Peres, arXiv:2608.23545, Theorem 1.1)

Proof mechanism

The passage time

1 · boundary routing

Florescu, Levine and Peres (2016) showed that, if \(T(n)<\infty\), then between \(T(n)\) and \(T(n+1)\) every directed edge is used at most once, and that, if \(T(n+1)<\infty\), the walk departs exactly \(\deg(x)\) times from every \(x\in A_n\). For a finite set \(S\) of vertices, place one particle at the head of each directed edge leaving \(S\), declare \(S\) absorbing, and move the particles by the rotor rule until every particle has stepped into \(S\). When this procedure terminates, let \(\Phi(S)\) be the union of \(S\) and the set of vertices visited during it. The map \(\Phi\) is monotone, and if \(T(n+1)<\infty\), then

\[A_{n+1}=\Phi(A_n),\qquad A_n=\Phi^n(\{o\}).\]

2 · passage time

The passage time, the number of circuits required for the walk started at \(x\) to reach \(y\), \[\tau(x,y)=\min\{n:y\in\Phi^n(\{x\})\}\] satisfies the triangle inequality, is stationary in law, and is at most the graph distance. The subadditive ergodic theorem provides a deterministic limit \(\tau(o,nz)/n\to\mu(z)\), and the unit ball of \(\mu\) is the limit shape \(B\). The almost-live-path estimate, stated below, shows that \(\mu\) does not vanish.

3 · circuit clock

Write \(d(S)=\sum_{x\in S}\deg(x)\). If \(T(n)<\infty\) for every \(n\), then, for every \(n\),

\[d(A_n)\le T(n+1)-T(n)\le d(A_{n+1}).\]

If, in addition, \(|A_n|=\alpha n^2+o(n^2)\) and \(d(A_n)=\beta n^2+o(n^2)\) for some \(\alpha,\beta>0\), then

\[T(n)=\frac{\beta}{3}n^3+o(n^3).\]

Moreover \(|R_t|t^{-2/3}\to\alpha(3/\beta)^{2/3}\), and if \(n^{-1}A_n\to B\) in Hausdorff distance, then \(t^{-1/3}R_t\to(3/\beta)^{1/3}B\).

Details

Live paths. A path \(x_0,x_1,\ldots\) of distinct vertices is live at an internal vertex \(x_i\) if, in the cyclic order beginning immediately after the initial rotor at \(x_i\), the edge \(x_i\to x_{i+1}\) occurs before \(x_i\to x_{i-1}\). If \(y\in\Phi^n(\{x\})\), then some path from \(x\) to \(y\) fails the live condition at no more than \(n-1\) of its internal vertices. The almost-live-path estimate states that, for some \(\eta>0\), the probability that some path starting with a given directed edge \(u\to v\) reaches graph distance \(R\) from \(u\) while the live condition fails at no more than \(\eta R\) internal vertices tends to zero as \(R\to\infty\), uniformly in \(u\to v\).

Square lattice. Rotating a directed edge \(v\to w\) counterclockwise through \(90^\circ\) about its midpoint gives a dual edge, called open when \(v\to w\) comes second or third in the cyclic order beginning after the initial rotor at \(v\). The four dual edges at a vertex form a square, of which two adjacent sides are open; no open dual path traverses a fixed five-edge pattern, and percolation avoiding that pattern is subcritical. The sides of a square are not independent, so the estimate is transferred to the rotor model by an exploration.

Displayed statistic. The last readout is the least-squares slope of \(\log |R_s|\) against \(\log s\) over geometrically spaced times \(s\in[t/10,t]\). By the theorem, \(|R_t|t^{-2/3}\) converges almost surely to a deterministic limit in \((0,\infty)\).

A. Bou-Rabee and Y. Peres, “Eulerian walkers on \(\mathbb{Z}^{2}\) have range exponent 2/3” (2026), arXiv:2608.23545. The model is due to V. B. Priezzhev, D. Dhar, A. Dhar and S. Krishnamurthy, Phys. Rev. Lett. 77 (1996), cond-mat/9611019. On the shape of the visited region, R. Kapri and D. Dhar, Phys. Rev. E 80 (2009), arXiv:0906.5506; on the range, L. Florescu, L. Levine and Y. Peres, Amer. Math. Monthly 123 (2016), arXiv:1408.5533.

Kozma’s centrally excited walk

Random walks with memory

Kozma’s centrally excited walk

A centrally excited random walk on \(\mathbb Z^d\) moves like simple random walk, except that its first step from each site other than the origin has a drift of fixed size toward the origin. After \(n\) steps the visited set approximates a ball with radius of order \(n^{1/(d+1)}\).

Definition. Let \(d\ge2\) and \(0<\kappa<1/d\). A centrally excited random walk \(X_0=0,X_1,X_2,\ldots\) on \(\mathbb Z^d\) moves like simple random walk, except that on its first departure from each site \(x\neq0\) it moves to \(x\pm e_i\) with probability \(\tfrac1{2d}\mp\tfrac{\kappa x_i}{2|x|}\) for \(1\le i\le d\), where \(e_1,\ldots,e_d\) is the standard basis of \(\mathbb R^d\). Given the path up to that departure, this step has mean \(-\kappa x/|x|\), a drift of size \(\kappa\) toward the origin. The local time \(\ell_n(x)\) is the number of departures from \(x\) before time \(n\), and the range \(A_n\) is the set of sites \(x\) with \(\ell_n(x)>0\).

Norm

Colour

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Steps \(n\): 0
Range size \(|A_n|\): 0
Radius \(r_n\): 0
Inner and outer radii ÷ \(r_n\): –

Arrows: mean of the first step from each site x ≠ 0 not in Aₙ · every other step is a step of simple random walk

One sample on \(\mathbb Z^2\) with \(\kappa=0.4\), up to 4,000,000 steps, with no boundary. After the first few hundred steps, the picture is rescaled by \(r_n=\bigl(3n/(4\kappa|B|)\bigr)^{1/3}\), where \(|B|\) is the area of the unit ball of the chosen norm; the outline is the ball of radius \(r_n\). For \(\ell^1\), \(x/|x|\) is replaced by the subgradient \(\xi(x)=(\operatorname{sign}x_1,\operatorname{sign}x_2)\) of the norm at \(x\); for \(\ell^\infty\), by \(\xi(x)=\operatorname{sign}(x_i)\,e_i\) when \(|x_i|>|x_{3-i}|\), and by \(\xi(x)=\tfrac12\bigl(\operatorname{sign}(x_1)\,e_1+\operatorname{sign}(x_2)\,e_2\bigr)\) when \(|x_1|=|x_2|\). First-departure time colours each site \(x\) by \((t_x/n)^{1/3}\), where \(t_x\) is the first-departure time of \(x\). Local time colours \(\ell_n(x)/(4\kappa r_n)\); the dashed curves are the level sets \(\tfrac14,\tfrac12,\tfrac34\) of \(1-\|x\|/r_n\), where \(\|x\|\) is the chosen norm. In the readout, the inner radius is the smallest norm of a site not in \(A_n\), and the outer radius is the largest norm of \(X_j\) for \(0\le j\le n\).

Theorem (Limit shape). Let \(d\ge2\) and \(0<\kappa<1/d\), and let \(r_n=\bigl((d+1)n/(2d\kappa\omega_d)\bigr)^{1/(d+1)}\), where \(\omega_d\) is the volume of the unit ball. Then, almost surely, the following hold. For every \(0<\eta<1\) and all sufficiently large \(n\), every \(x\in\mathbb Z^d\) with \(|x|<(1-\eta)r_n\) lies in \(A_n\), and every \(x\in A_n\) satisfies \(|x|<(1+\eta)r_n\). As \(n\to\infty\), \[\frac1{r_n}\max_{x\in\mathbb Z^d}\bigl|\ell_n(x)-2d\kappa(r_n-|x|)_+\bigr|\longrightarrow0.\] The walk visits every site of \(\mathbb Z^d\) infinitely often. (Bou-Rabee–Peres, Theorem 1.1)

Other norms, and the size of the fluctuations

Replace \(x/|x|\) in the first-departure probabilities by a fixed subgradient \(\xi(x)\) of a norm \(\Psi\) at \(x\), and assume \(\kappa\max_i\Psi(e_i)<1/d\). Then the range approximates \(\{\Psi<r_n\}\) and the local times approximate \(2d\kappa(r_n-\Psi)_+\), with the volume of the unit ball of \(\Psi\) in place of \(\omega_d\). The theorem includes norms with corners, such as \(\|\cdot\|_1\) and \(\|\cdot\|_\infty\). (Theorem 2.1)

For the Euclidean norm in the plane, for every \(p>0\) there is \(C\) such that, for every \(n\ge2\), with probability at least \(1-Cn^{-p}\) the inner radius differs from \(r_n\) by at most \(C\sqrt{r_n\log n}\) and the outer radius exceeds \(r_n\) by at most \(C\sqrt{r_n}(\log n)^{5/2}\). For every \(p>0\) there is \(c>0\) such that, for all sufficiently large \(n\), the inner radius is at most \(r_n-c\sqrt{r_n\log n}\) with probability at least \(n^{-p}\), and the outer radius is at least \(r_n+c\sqrt{r_n\log n}\) with probability at least \(n^{-p}\). Here the inner radius is the distance from the origin to the complement of the union of the unit cells of \(A_n\); the inner radius in the readout differs from it by less than 1. Since \(r_n\) is of order \(n^{1/3}\), the radii fluctuate on the scale \(n^{1/6}\) up to logarithmic factors. (Theorems 1.2 and 1.3)

A. Bou-Rabee and Y. Peres, “Kozma’s centrally excited walk converges to a Euclidean ball” (2026), arXiv:2610.10162. Lean formalization.

Diffusion-limited aggregation

Related model

Diffusion-limited aggregation

Definition. Let \(A_1=\{0\}\subset\mathbb Z^2\). Conditional on \(A_n\), sample \(Z_{n+1}\) from harmonic measure viewed from infinity on the outer vertex boundary of \(A_n\), and set \(A_{n+1}=A_n\cup\{Z_{n+1}\}\).

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Loading run

Particles: 0
Maximum Euclidean radius \(R_k\): 0
Least-squares slope of \(\log|A_k\cap B_r|\) against \(\log r\): —
View width: 0

Attachment order · logarithmic

Diffusion-limited aggregation in a square of side N = 2,201, replayed from a stored run of 60,000 particles. Before particle k, let R be the maximum Euclidean radius of the cluster: choose θ uniformly, start at the lattice point nearest (R + 4)(cos θ, sin θ), and run simple random walk until it reaches an unoccupied site adjacent to the cluster; if its distance from the origin exceeds min{2.2(R + 4) + 20, N/2 − 3}, restart it with a new θ. Colour: attachment order on a logarithmic scale; bright tips mark recent attachments. The ring marks the seed.

Observation. For the displayed 60,000-particle square-lattice sample, with \(B_r=\{x\in\mathbb Z^2:|x|_2<r\}\) and \(M(r)=|A_{60000}\cap B_r|\), ordinary least-squares regression of \(\log M(r)\) on \(\log r\) gives slope 1.695219.

Question. Determine the growth rate of the radius of \(A_n\); Kesten proved that it is at most of order \(n^{2/3}\).

Details

The regression uses 26 logarithmically spaced radii \(6\le r\le0.85R_{60000}\), retaining radii with \(M(r)>12\); its root-mean-square log residual is 0.068296.

T. A. Witten and L. M. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon”, Phys. Rev. Lett. 47 (1981) 1400. For rigorous upper bounds on the growth rate, H. Kesten, “How long are the arms in DLA?”, J. Phys. A 20 (1987). Y. L. Loh, “Bias-free simulation of diffusion-limited aggregation on a square lattice”, arXiv:1407.2586, verified that the fractal dimension of square-lattice DLA evolves from the continuum DLA value 1.71 for small disk-shaped clusters towards Kesten’s bound 3/2 for highly anisotropic clusters. D. S. Grebenkov and D. Beliaev observed that DLA clusters on the square lattice undergo strongly anisotropic growth, with the maximal growth rate along the axes: “How anisotropy beats fractality in two-dimensional on-lattice DLA growth”, Phys. Rev. E 96 (2017), arXiv:1708.01514.

Internal DLA

Related model

Internal DLA

Definition. Fix \(d\ge2\) and set \(A_0=\varnothing\subset\mathbb Z^d\). Given independent simple random walks \(S^{(n)}\) on \(\mathbb Z^d\), each started at the origin, let \(\tau_n=\inf\{k\ge0:S_k^{(n)}\notin A_{n-1}\}\) and \(A_n=A_{n-1}\cup\{S_{\tau_n}^{(n)}\}\).

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Walk

Particles: 1
Walk steps: 0
Outradius − inradius: 0
Outradius: 0

Settlement order \(k\), in bands of \(\sqrt{k/\pi}\)

Internal DLA on a 141-by-141 grid: independent simple random walks start at the origin and settle at the first unoccupied site. Colour: settlement order \(k\), in bands of equal width in \(\sqrt{k/\pi}\). Readouts: r_out = max{|x| : x occupied} and r_out − r_in with r_in = min{|x| : x unoccupied}. The run stops when the cluster reaches the boundary of the grid.

Theorem. Fix \(d\ge2\). Let \(\omega_d\) be the volume of the Euclidean unit ball, put \(B_r=\{x\in\mathbb Z^d:|x|_2<r\}\), and set \(r_n=(n/\omega_d)^{1/d}\). Then \(r_n^{-1}A_n\) converges almost surely in Hausdorff distance to the unit Euclidean ball. (Lawler–Bramson–Griffeath, Ann. Probab. 20 (1992))

Theorem. For each fixed \(d\ge2\), there exists a deterministic \(C=C(d)<\infty\) such that, almost surely, for all sufficiently large \(r\), \[B_{r-C\log r}\subset A_{\lfloor\omega_2r^2\rfloor}\subset B_{r+C\log r}\qquad(d=2),\] and \[B_{r-C\sqrt{\log r}}\subset A_{\lfloor\omega_dr^d\rfloor}\subset B_{r+C\sqrt{\log r}}\qquad(d\ge3).\] In continuous-time planar IDLA, let \(T(t)\) be an independent rate-one Poisson process, let \(F(x)=\inf\{t:x\in A_{T(t)}\}\), \(L(x)=\sqrt{F(x)/\pi}-|x|\), and \(G_R(z)=L(\lfloor Rz\rfloor)\). For a test function \(\phi\), define \((\phi,G_R)=\int_{\mathbb R^2}\phi(z)G_R(z)\,dz\). For every finite collection \(\phi_1,\ldots,\phi_m\) in \(H_0=\operatorname{span}\{\operatorname{Re}(az^k)f(|z|):a\in\mathbb C,\ k\ge0,\ f\in C_c^\infty((0,\infty))\}\), the vector \(((\phi_j,G_R))_{j=1}^m\) converges in distribution as \(R\to\infty\) to the corresponding vector of pairings with the planar augmented Gaussian free field. (Jerison–Levine–Sheffield, arXiv:1010.2483, arXiv:1012.3453 and arXiv:1101.0596)

Question. Is the logarithmic bound on the maximal fluctuations sharp in dimension \(d=2\)?

Details

Proof idea. The connection between IDLA and the divisible sandpile was implicit in the first work of Lawler, Bramson and Griffeath: a lower bound on the divisible sandpile cluster implies the same lower bound for the IDLA cluster. The logarithmic maximal fluctuation bounds use martingales built from discrete harmonic functions, an estimate excluding thin tentacles, and an argument controlling early and late points. For the fluctuation theorem, sums of discrete harmonic polynomials over the cluster are martingales; their quadratic variations converge, and a martingale central limit theorem gives the augmented Gaussian free field.

P. Meakin and J. M. Deutch, J. Chem. Phys. 85 (1986), and P. Diaconis and W. Fulton, Rend. Sem. Mat. Univ. Politec. Torino 49 (1991). G. Lawler, M. Bramson and D. Griffeath, “Internal diffusion limited aggregation”, Ann. Probab. 20 (1992); G. Lawler, “Subdiffusive fluctuations for internal DLA”, Ann. Probab. 23 (1995). D. Jerison, L. Levine and S. Sheffield, J. Amer. Math. Soc. 25 (2012), arXiv:1010.2483, Electron. J. Probab. 18 (2013), arXiv:1012.3453, and Duke Math. J. 163 (2014), arXiv:1101.0596; fluctuation results were obtained independently by A. Asselah and A. Gaudillière.

Uniform spanning tree

Related model

Uniform spanning tree

Definition. On a finite connected graph, Wilson’s algorithm maintains a current tree, which initially consists of just the root. While there remain vertices not in the tree, it runs a simple random walk from the first such vertex in a fixed order of the vertices, erasing cycles as they are created, until the walk encounters the current tree; then the cycle-erased trajectory is added to the current tree.

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Root

Random-walk steps: 0
Tree edges: 0
Erased edges: 0
Edges of the current trajectory: 0
Leaves · share of tree vertices: 0

Order of addition: earlylate

Wilson’s algorithm on the 96-by-96 grid, rooted at one of the four vertices nearest its centre; the walks start from the vertices not in the tree in order of distance from the root. Cyan: the current cycle-erased trajectory. Coral: the cycle being erased. Yellow: the vertex at which the walk encounters the current tree. Tree edges are coloured by the order in which they were added. The finished tree has 9,215 edges.

Theorem. On a finite connected graph, for every root and every order of the vertices, Wilson’s algorithm returns a uniformly random spanning tree. (Wilson, 1996, Theorems 1 and 4)

Theorem. Let \(D\subsetneq\mathbb C\) be a simply connected domain with \(0\in D\). For \(\delta>0\), let \(\mu_\delta\) be the law of the loop-erasure of simple random walk on the grid \(\delta\mathbb Z^2\), started at 0 and stopped when it hits \(\partial D\), and let \(\nu\) be the law of the image of the radial \(\mathrm{SLE}_2\) path under a conformal map from the unit disk to \(D\) fixing 0. Then \(\mu_\delta\) converges weakly to \(\nu\) as \(\delta\to0\), with respect to the metric \(\rho(\beta,\gamma)=\inf\sup_{t\in[0,1]}|\hat\beta(t)-\hat\gamma(t)|\) on unparameterized paths, where the infimum is over all parameterizations \(\hat\beta,\hat\gamma\) of \(\beta,\gamma\) by \([0,1]\). (Lawler–Schramm–Werner, arXiv:math/0112234, Theorem 1.1)

Theorem. Let \(D\subset\mathbb C\) be a domain whose boundary is a \(C^1\)-smooth simple closed curve, let \(a,b\in\partial D\) be distinct, and let \(\alpha,\beta\) be the two complementary arcs of \(\partial D\) whose endpoints are \(a\) and \(b\). For \(\delta>0\), let \(\gamma_\delta\) be the Peano curve of the UST on an approximation of \(D\) in the grid \(\delta\mathbb Z^2\), with wired boundary near \(\alpha\) and free boundary near \(\beta\). Then \(\gamma_\delta\) converges weakly, with respect to \(\rho\), to the image of the chordal \(\mathrm{SLE}_8\) path under a conformal map from the upper half-plane to \(D\) mapping 0 to \(a\) and \(\infty\) to \(b\). (Lawler–Schramm–Werner, arXiv:math/0112234, Theorem 1.3)

Details

Proof idea. Wilson associates with each vertex other than the root an infinite stack of independent random successors and pops directed cycles off the stacks; the choice of which cycle to pop next is irrelevant, and the probability that the stacks define a tree \(T\) and a set of popped cycles factors into a term depending on the cycles alone and the weight of \(T\). Lawler, Schramm and Werner (2004) first show that the Loewner driving process of the discrete path converges to Brownian motion, using a Markovian property of the path and a martingale observable: for the loop-erased walk, the expected number of visits to a vertex \(v\) by the simple random walk generating it; for the Peano curve, the probability that the path in the UST from \(v\) to \(\alpha\) hits a given subarc of \(\alpha\). Regularity properties of the discrete paths then give convergence in the metric \(\rho\).

D. B. Wilson, “Generating random spanning trees more quickly than the cover time”, STOC (1996). G. Lawler, O. Schramm and W. Werner, “Conformal invariance of planar loop-erased random walks and uniform spanning trees”, Ann. Probab. 32 (2004), arXiv:math/0112234. The loop-erased walk is due to G. Lawler (1980). A. Broder (1989) and D. Aldous (1990) found an earlier random-walk algorithm for uniform spanning trees.

Rotor-router aggregation

Related model

Rotor-router aggregation

Definition. Fix \(d\ge2\), a cyclic order of \(\{\pm e_1,\ldots,\pm e_d\}\), and an initial rotor at every site of \(\mathbb Z^d\). Set \(A_0=\varnothing\). For \(n\ge1\), release a particle at the origin; while it is in \(A_{n-1}\), the rotor at its current location is rotated to the next direction in the cyclic order, and the particle takes a step in the direction of the newly rotated rotor. At the first site \(z_n\notin A_{n-1}\), stop the particle and set \(A_n=A_{n-1}\cup\{z_n\}\). The displayed run has \(d=2\), clockwise rotor order, and all rotors initially east.

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Particles: 1
Steps: 0
Sites whose rotor has been rotated: 0
Outradius: 0
Outradius − inradius: 0

Rotor direction

Rotor-router aggregation on a 141-by-141 grid; every rotor initially points east. Each particle starts at the origin; at an occupied site the rotor is rotated clockwise by 90 degrees and the particle takes a step in the direction of the newly rotated rotor; it stops at the first unoccupied site. Each site is colored according to the direction of its rotor. Readouts: r_out = max{|x| : x occupied} and r_out − r_in with r_in = min{|x| : x unoccupied}. The run stops when r_out reaches the boundary of the grid.

Theorem. Let \(A_n\) be rotor-router aggregation on \(\mathbb Z^d\), started with \(n\) particles at the origin and any initial rotor configuration. Let \(\omega_d\) be the volume of the Euclidean unit ball, put \(B_s=\{x\in\mathbb Z^d:|x|_2<s\}\), and set \(r=(n/\omega_d)^{1/d}\). There are constants \(c_1,c_2\), depending only on \(d\), such that \[B_{r-c_1\log r}\subset A_n\subset B_{r+c_2r^{1-1/d}\log r}.\] (Levine–Peres, arXiv:0704.0688, Theorem 1.1)

Observation. Take one simulated cluster of 1,200,000 particles for each of rotor-router aggregation and internal DLA in \(\mathbb Z^2\), so that \(r=(n/\pi)^{1/2}=618.04\). Divide the plane into 720 equal angular sectors about the origin, and in each sector let \(r(\theta)\) be the largest distance from the origin of an occupied site. Over the 720 sectors, \(r(\theta)\) has standard deviation 0.53 and range 2.35 for rotor-router aggregation, and 0.98 and 6.58 for internal DLA; for the lattice disk \(\{x:|x|\le r\}\) the values are 0.16 and 0.83.

Question. Do there exist constants \(c,c'\) depending only on \(d\) such that \(B_{r-c}\subset A_n\subset B_{r+c'}\) for every \(n\)?

Details

Proof idea. Let \(u(x)\) be the total number of exits from \(x\) by the first \(n\) particles. Its gradient along an edge differs from \(-2d\) times the net number of crossings of that edge by at most \(4d-2\), and estimates on the Green's function for simple random walk stopped on exiting \(B_r\) then give \(u>0\) on \(B_{r-c\log r}\). For the outer bound, each particle is stopped when it reaches an unoccupied site or the spherical shell \(\mathcal S_\rho\) of radius \(\rho\); the particles stopped on the shell are restarted using the abelian property, and when \(h\) is of order \(r^{1-1/d}\) a constant fraction of the particles that reach \(\mathcal S_\rho\) find unoccupied sites before reaching \(\mathcal S_{\rho+h}\).

Rotor-router walk is a deterministic analogue of random walk, first studied by Priezzhev, Dhar, Dhar and Krishnamurthy (1996) under the name “Eulerian walkers”; rotor-router aggregation was introduced by Propp. Internal DLA is the analogous growth model defined using random walks instead of rotor-router walks.

L. Levine and Y. Peres, “Spherical asymptotics for the rotor-router model”, Indiana Univ. Math. J. 57 (2008), arXiv:math/0503251; “Strong spherical asymptotics…”, Potential Anal. 30 (2009), arXiv:0704.0688; and the survey “Laplacian growth, sandpiles and scaling limits”, Bull. Amer. Math. Soc. 54 (2017), arXiv:1611.00411. Propp’s model is described by L. Levine, arXiv:math/0409407.

Integer superharmonic matrices on \(\mathbb{Z}^{2}\)

Related model

Integer superharmonic matrices on \(\mathbb{Z}^{2}\)

Definition. For \(k\in\mathbb Z\), let \(\mathcal B_k\) be the Apollonian circle packing generated by the vertical lines through \((2k,0)\) and \((2k+2,0)\) together with the circle of radius 1 centred at \((2k+1,0)\): the minimal collection of general circles (circles or lines) containing these three that is closed under the addition of Soddy general circles. The band packing is \(\mathcal B=\bigcup_{k\in\mathbb Z}\mathcal B_k\). With \(\Delta g(x)=\sum_{y\sim x}(g(y)-g(x))\), a symmetric matrix \(A\) is integer superharmonic when an integer-valued \(g\) satisfies \(\Delta g\le1\) and \(g(x)=\tfrac12x\cdot Ax+o(|x|^2)\). Write \(A\preccurlyeq B\) when \(B-A\) is positive semidefinite. If \(C\in\mathcal B\) has centre \((x_1,x_2)\) and radius \(r\), then \(A_C=\tfrac12\bigl(\begin{smallmatrix}r+x_1&x_2\\x_2&r-x_1\end{smallmatrix}\bigr)\).

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Run

Parent circles

Depth: 1 / 4
Child curvature \(1/r\): 4
Parent curvatures: 0 · 1 · 1
A parent is a line: yes

Circlesparents child

The band packing \(\mathcal B_0\), rotated by 90 degrees: two parallel lines at distance 2, the circles of radius 1 tangent to both, and, recursively, the Soddy circles of every pairwise tangent triple. Tour: four children, of curvature 4, 9, 28 and 81, each with its three parents. Depth is the number of Soddy circle additions from the circles of radius 1. Explore: every circle of curvature at most 500 in 12 periods of the packing; drag to pan, scroll to zoom.

Theorem. Let \(\mathcal B\) be the Apollonian band packing defined above. A symmetric matrix \(A\) is integer superharmonic on \(\mathbb{Z}^{2}\) precisely when \(\operatorname{tr}A\le0\), or \(A\preccurlyeq A_C\) for some circle \(C\in\mathcal B\). If \(C\) has centre \((x_1,x_2)\) and radius \(r\), then \[A_C=\frac12\begin{pmatrix}r+x_1&x_2\\x_2&r-x_1\end{pmatrix}.\] For each circle \(C\in\mathcal B\), with \(L_C=\{v\in\mathbb Z^2:A_Cv\in\mathbb Z^2\}\), there is an integer superharmonic representative \(g_C\) for \(A_C\) with \(g_C(x+v)=g_C(x)+x^{\mathsf T}A_Cv+g_C(v)\) for all \(v\in L_C\) and \(x\in\mathbb Z^2\), which is maximal in the sense that \(g-g_C\) is bounded whenever \(g:\mathbb Z^2\to\mathbb Z\) satisfies \(\Delta g\le1\) and \(g\ge g_C\); and there is a tile \(T_C\) with \(90^\circ\) rotational symmetry such that \(T_C+L_C\) is a tiling of \(\mathbb Z^2\). The curvature of \(C\) is the area of \(T_C\). (Levine–Pegden–Smart, arXiv:1309.3267, Theorems 1.1–1.3)

Question. Describe the set of integer superharmonic matrices for periodically embedded graphs other than \(\mathbb{Z}^{2}\).

Details

Proof idea. For each circle \(C\in\mathcal B\), construct a tile \(T_C\) that tiles the plane periodically and decomposes into copies of the tiles of the parent circles of \(C\) with specified overlaps. Use the decomposition of \(T_C\) to define recursively a tile odometer on \(T_C\). Extend the tile odometer to \(\mathbb Z^2\) via the periodicity condition, and check that the extension \(g_C\) satisfies \(\Delta g_C\le1\) and is maximal.

The condition \(\operatorname{tr}A\le0\) in the theorem adds a Cantor set of matrices in the plane \(\operatorname{tr}A=0\) that lie in the topological closure of the downset of \(\{A_C:C\in\mathcal B\}\) but not in the downset itself.

L. Levine, W. Pegden and C. K. Smart, “Apollonian structure in the Abelian sandpile”, Geom. Funct. Anal. 26 (2016), arXiv:1208.4839; and “The Apollonian structure of integer superharmonic matrices”, Ann. of Math. 186 (2017), arXiv:1309.3267. For the packings themselves, R. Graham, J. Lagarias, C. Mallows, A. Wilks and C. Yan, “Apollonian circle packings: number theory”, arXiv:math/0009113, and “Geometry and group theory I: the Apollonian group”, arXiv:math/0010298; on counting, A. Kontorovich and H. Oh, J. Amer. Math. Soc. 24 (2011), arXiv:0811.2236.

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