Report

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 1 2 4 3 2 3 4 1 3 4 1 2 1 Groves: graph with marked nodes 3 5 4 2 1 3 5 1 3 5 4 1 3 4 2 4 3 5 2 2 5 1 2 1 3 5 4 2 4 1 3 5 4 2 Uniformly random grove 1 3 5 4 2 Goal: compute ratios of partition functions in terms of electrical quantities Kirchhoff’s formula for resistance 1 3 5 Arbitrary finite graph with two special nodes 4 2 1 3 5 1 4 2 3 5 1 4 3 5 1 4 3 5 4 3 5 4 2 1 3 5 4 2 3 spanning trees 3 4 2 4 2 1 5 3 5 2 2 2 5 2-tree forests with nodes 1 and 2 separated 1 1 1 3 5 4 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 1 1 1 4 3 4 5 5 4 2 2 circular planar circular planar 3 2 planar, not circular planar Planar graph Special vertices called nodes on outer face Nodes numbered in counterclockwise order along outer face 3 1 2 4 3 2 3 4 3 4 1 1 2