Local statistics of
the abelian sandpile model
David B. Wilson
Key ingredients
• Bijection between ASM’s and spanning trees:
Dhar
Majumdar—Dhar
Cori—Le Borgne
Bernardi
Athreya—Jarai
• Basic properties of spanning trees
Pemantle
Benjamini—Lyons—Peres—Schramm
• Computation of topologically defined events for
spanning trees
Kenyon—Wilson
[sandpile demo]
Infinite volume limit
• Infinite volume limit exists (Athreya—Jarai ’04)
• Pr[h=0]= 2=¼2 ¡ 4=¼3 (Majumdar—Dhar ’91)
• Other one-site probabilities computed by
Priezzhev (’93)
Priezzhev (’94)
Jeng—Piroux—Ruelle (’06)
[burning bijection demo]
Underlying graph
Uniform spanning tree
Uniform spanning tree on infinite grid
Pemantle: limit of UST on large boxes
converges as boxes tend to Z^d
Pemantle: limiting process has one tree
if d<=4, infinitely many trees if d>4
Uniform spanning tree
UST and LERW on Z^2
Benjamini-Lyons-Peres-Schramm:
UST on Z^d has one end if d>1,
i.e., one path to infinity
Local statistics of UST
Local statistics of UST can be
computed via determinants
of transfer impedance matrices
(Burton—Pemantle)
Why doesn’t this give local
statistics of sandpiles?
Sandpile density and LERW
Conjecture: path to infinity visits
neighbor to right
with probability 5/16
(Levine—Peres,
Poghosyan—Priezzhev)
Sandpile density and LERW
Theorem: path to infinity visits
neighbor to right
with probability 5/16
(Poghosyan-Priezzhev-Ruelle,
Kenyon-W)
JPR integral evaluates to ½
(Caracciolo—Sportiello)
Kenyon—W
Kenyon—W
Kenyon—W
Kenyon—W
Joint distribution of heights
at two neighboring vertices
Higher dimensional marginals of sandpile heights
Pr[3,2,1,0 in 4x1 rectangle] =
Sandpiles on hexagonal lattice
(One-site probabilities also computed by Ruelle)
Sandpiles on triangular lattice
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2
4
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3
4
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3
4
1
2
1
Groves: graph with marked nodes
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5
4
2
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Uniformly random grove
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3
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2
Goal: compute ratios of partition functions
in terms of electrical quantities
Kirchhoff’s formula for resistance
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3
5
Arbitrary finite graph with two special nodes
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2
1
3
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3 spanning trees
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4
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2
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5
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2
5 2-tree forests with nodes 1 and 2 separated
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1
1
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2
Arbitrary finite graph with two special nodes
three
(Kirchoff)
Arbitrary finite graph with four special nodes?
1
4
All pairwise resistances are equal
5
2
3
1
4
2
3
All pairwise resistances are equal
Need more than boundary measurements (pairwise resistances)
Need information about internal structure of graph
Circular planar graphs
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1
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4
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circular planar
circular planar
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2
planar,
not circular planar
Planar graph
Special vertices called nodes on outer face
Nodes numbered in counterclockwise order along outer face
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1
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