A Structure Theory of the Sandpile Monoid for Directed Graphs
Local statistics of the abelian sandpile model David B. Wilson TexPoint fonts used in EMF. Read the...
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Transcript of Local statistics of the abelian sandpile model David B. Wilson TexPoint fonts used in EMF. Read the...
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Local statistics ofthe abelian sandpile model
David B. Wilson
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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
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[sandpile demo]
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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
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Priezzhev (’94)
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Jeng—Piroux—Ruelle (’06)
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[burning bijection demo]
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Underlying graph Uniform spanning tree
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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
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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
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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?
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Sandpile density and LERW
Conjecture: path to infinity visitsneighbor to rightwith probability 5/16(Levine—Peres, Poghosyan—Priezzhev)
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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)
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Kenyon—W
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Kenyon—W
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Kenyon—W
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Kenyon—W
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Joint distribution of heightsat two neighboring vertices
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Higher dimensional marginals of sandpile heights
Pr[3,2,1,0 in 4x1 rectangle] =
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Sandpiles on hexagonal lattice
(One-site probabilities also computed by Ruelle)
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Sandpiles on triangular lattice
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Groves: graph with marked nodes
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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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5 4
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1 3Arbitrary finite graph with two special nodes
(Kirchoff)
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three
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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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