Local statistics of the abelian sandpile model David B. Wilson TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A.

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Presentation transcript:

Local statistics of the abelian sandpile model David B. Wilson TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A

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]= (Majumdar—Dhar ’91) Other one-site probabilities computed by Priezzhev (’93)

Priezzhev (’94)

Jeng—Piroux—Ruelle (’06)

[burning bijection demo]

Underlying graphUniform 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

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

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

Groves: graph with marked nodes

Uniformly random grove

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

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

Arbitrary finite graph with two special nodes (Kirchoff) 3 three

Arbitrary finite graph with four special nodes? All pairwise resistances are equal Need more than boundary measurements (pairwise resistances) Need information about internal structure of graph

Planar graph Special vertices called nodes on outer face Nodes numbered in counterclockwise order along outer face Circular planar graphs circular planar planar, not circular planar