This document gives an introduction to computing and visualizing various Abelian networks in Julia.

For a tutorial on installing Julia and setting up Jupyter notebooks see here. For a general introduction to Julia from the viewpoint of Linear Algebra, check out this page.

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using Plots # for visualization to install type in ] add Plots


# Abelian Sandpile¶

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#######################
## Description
##		computes the stabilization of the 2-dimensional
##		sandpile on a square
## Inputs
## 		S: an array of integers specifying the initial condition
## Output
##      returns odometer, T, sandpile S is stabilized in place

function stabilize!(S)
#initialize the all 0 odometer
T = fill(0, size(S))

#iterate over all sites and try to topple
#until it is not possible anymore
isTopple = true;
while(isTopple)
isTopple = false
for y in 1:size(S,2)
for x in 1:size(S,1)
isTopple = isTopple | topple!(S,T,x,y)
end
end
end

return T;
end

## fire a single site in S once
## i.e. increment odometer
## remove chips from S[x,y]
## and give chips to neighbors
## (x+1,y), (x-1,y), (x,y+1), (x,y-1)
function topple!(S,T,x,y)
#number of times fire site (x,y)
#we take max in case there is a hole at (x,y)
z = max(floor(S[x,y]/4), 0)
#increment the odometer
T[x,y]+=z
S[x,y]-=4*z;

#give sand to each neighbor
if(x > 1)
S[x-1,y]+=z
end
if(x < size(S,1))
S[x+1,y]+=z
end
if(y > 1)
S[x,y-1]+=z
end
if(y < size(S,2))
S[x,y+1]+=z
end

#if we have toppled return true
#else return false
if(z > 0)
return true;
else
return false
end
end

Out:
topple! (generic function with 2 methods)
##### Abelian sandpile examples¶

We compute several examples of Abelian sandpiles using the above function.

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N = 50
S = fill(4,N,N)
T = stabilize!(S)
heatmap(S)

Out:
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N = 200
S = fill(1,N,N)
S[div(N,2),div(N,2)]+=N^2
T = stabilize!(S); #will take a couple of seconds
S[T.==0].=-1;
heatmap(S)

Out:
In :
N = 200
S = rand(0:1,N,N)
init_condition = copy(S)
S[div(N,2),div(N,2)]+=N^2
T = stabilize!(S) #will take a couple of seconds
S = S .- init_condition
S[T.==0].=-1;
heatmap(S)

Out:
##### Efficient computation of symmetric Abelian sandpiles in arbitrary dimensions¶

If you have access to a GPU and want to compute large symmetric (e.g. single-source, constant, ...) sandpiles in $\mathbf{Z}^d$ see code here. This code uses symmetry of the initial condition to reduce to computing the sandpile on the simplex, $\mathcal{S}_M = \{ (x_1, \ldots, x_d) \in \mathbf{Z}^d | M \leq x_1 \leq \cdots \leq x_d \leq 1 \}$. In high dimensions, computing sandpiles on the simplex improves space complexity by a factor of $d^d$. This reduction in size also leads to a faster algorithm when using parallelization.

For example, an NVDIA RTX 2080 TI can compute the sandpile on $\mathbf{Z}^4$ started with $2^{30}$ chips at the origin in about a day: ## Exploding sandpiles¶

We visualize exploding sandpiles via the parallel toppling procedure. First we modify the above code to perform parallel updates, then look at two examples of parallel toppling sandpiles.

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#######################
## Description
##		computes num_iter parallel topple updates of a sandpile on a square
## Inputs
## 		S: an array of integers specifying the sandpile
## 		T: an array of integers specifying current odometer
## 		num_iters: an integer specifying how many parallel update steps to take
## Output
##      sandpile S is updated in place

function parallel_topple!(S, T, num_iters)

for i in 1:num_iters
S_prev = copy(S)
for y in 1:size(S,2)
for x in 1:size(S,1)
one_topple!(S,S_prev,T,x,y)
end
end
end
end

## fire a single site in S once
## via the parallel update procedure
function one_topple!(S,S_prev, T,x,y)
## now topple according to how many
## grains it had at the PREVIOUS
## time step
z = max(floor(S_prev[x,y]/4), 0)

#number of times fire site (x,y)
#we take max in case there is a hole at (x,y)
#increment the odometer
T[x,y]+=z
S[x,y]-=4*z;

#give sand to each neighbor
if(x > 1)
S[x-1,y]+=z
end
if(x < size(S,1))
S[x+1,y]+=z
end
if(y > 1)
S[x,y-1]+=z
end
if(y < size(S,2))
S[x,y+1]+=z
end
end

Out:
one_topple! (generic function with 1 method)
In :
## start with all 4s everywhere in a square of side length 200
## and parallel topple for (25)^2 time steps
N = 100
S = fill(4,N,N)
T = fill(0,N,N)
parallel_topple!(S, T, (25)^2)
heatmap(S)

Out:
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## start with a checkerboard 2,3 in a square
## and parallel topple for (25)^2 time steps
N = 200
S = [2+(x+y)%2 for x in 1:N, y in 1:N]
T = fill(0,N,N)
#initial checkerboard
heatmap(S)

Out:
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#place M grains at the origin and parallel topple for M steps