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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
4
2
1
3
4
2
1
3
4
21
3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
Groves: graph with marked nodes
Uniformly random grove
5 4
2
1 3
Goal: compute ratios of partition functions in terms of electrical quantities
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
5 4
2
1 3
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
5 4
2
1 3Arbitrary finite graph with two special nodes
(Kirchoff)
3
three
Arbitrary finite graph with four special nodes?
5
32
1 4 All pairwise resistances are equal
32
1 4 All pairwise resistances are equal
Need more than boundary measurements (pairwise resistances)Need information about internal structure of graph
5 4
2
1 3
Planar graphSpecial vertices called nodes on outer faceNodes numbered in counterclockwise order along outer face
Circular planar graphs
5
32
1 4
circular planar circular planar
3
2
1
4
planar,not circular planar
4
21
3
4
2
1
3 4
21
3