Local statistics ofthe abelian sandpile model

David B. Wilson

Key ingredients

• Bijection between ASM’s and spanning trees:DharMajumdar—DharCori—Le BorgneBernardiAthreya—Jarai

• Basic properties of spanning treesPemantleBenjamini—Lyons—Peres—Schramm

• Computation of topologically defined events for spanning treesKenyon—Wilson

[sandpile demo]

Infinite volume limit• Infinite volume limit exists (Athreya—Jarai ’04)• Pr[h=0]= (Majumdar—Dhar ’91)• Other one-site probabilities computed by

Priezzhev (’93)

2=¼2 ¡ 4=¼3

Priezzhev (’94)

Jeng—Piroux—Ruelle (’06)

[burning bijection demo]

Underlying graph Uniform spanning tree

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

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 becomputed via determinantsof transfer impedance matrices (Burton—Pemantle)

Why doesn’t this give localstatistics of sandpiles?

Sandpile density and LERW

Conjecture: path to infinity visitsneighbor to rightwith probability 5/16(Levine—Peres, Poghosyan—Priezzhev)

Sandpile density and LERW

Theorem: path to infinity visitsneighbor to rightwith 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 heightsat 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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Groves: graph with marked nodes

Uniformly random grove

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Goal: compute ratios of partition functions in terms of electrical quantities

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Arbitrary finite graph with two special nodes

Kirchhoff’s formula for resistance

3 spanning trees

5 2-tree forests with nodes 1 and 2 separated

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1 3Arbitrary finite graph with two special nodes

(Kirchoff)

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Arbitrary finite graph with four special nodes?

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1 4 All pairwise resistances are equal

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1 4 All pairwise resistances are equal

Need more than boundary measurements (pairwise resistances)Need information about internal structure of graph

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Planar graphSpecial vertices called nodes on outer faceNodes numbered in counterclockwise order along outer face

Circular planar graphs

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circular planar circular planar

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planar,not circular planar

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