Local statistics of the abelian sandpile model

1. Local statistics ofthe abeliansandpile model David B. Wilson TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

2. 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

3. [sandpile demo]

5. Infinite volume limit • Infinite volume limit exists (Athreya—Jarai ’04) • Pr[h=0]= (Majumdar—Dhar ’91) • Other one-site probabilities computed by Priezzhev (’93)

6. Priezzhev (’94)

7. Jeng—Piroux—Ruelle (’06)

8. [burning bijection demo]

9. Underlying graph Uniform spanning tree

10. 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

11. 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

12. Local statistics of UST Local statistics of UST can be computed via determinantsof transfer impedance matrices (Burton—Pemantle) Why doesn’t this give localstatistics of sandpiles?

16. Sandpile density and LERW Conjecture: path to infinity visits neighbor to right with probability 5/16 (Levine—Peres, Poghosyan—Priezzhev)

17. 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)

18. Kenyon—W

19. Kenyon—W

20. Kenyon—W

21. Kenyon—W

22. Joint distribution of heights at two neighboring vertices

23. Higher dimensional marginals of sandpile heights Pr[3,2,1,0 in 4x1 rectangle] =

24. Sandpiles on hexagonal lattice (One-site probabilities also computed by Ruelle)

25. Sandpiles on triangular lattice

29. 1 2 4 3

30. 2 3 4 1

31. 3 4 1 2

32. 1 3 Groves: graph with marked nodes 5 4 2 1 3 1 3 1 3 5 4 5 4 5 4 2 2 2 1 3 1 3 1 3 5 4 5 4 5 4 2 2 2

33. Uniformly random grove

34. 1 3 5 4 2 Goal: compute ratios of partition functions in terms of electrical quantities

35. Kirchhoff’s formula for resistance 1 3 Arbitrary finite graph with two special nodes 5 4 2 1 3 1 3 1 3 1 3 1 3 5 4 5 4 5 4 5 4 5 4 2 2 2 2 2 5 2-tree forests with nodes 1 and 2 separated 1 3 1 3 1 3 5 4 5 4 5 4 2 2 2 3 spanning trees

36. 1 3 3 Arbitrary finite graph with two special nodes three 5 4 (Kirchoff) 2

37. Arbitrary finite graph with four special nodes? 1 4 All pairwise resistances are equal 5 2 3 1 4 All pairwise resistances are equal 3 2 Need more than boundary measurements (pairwise resistances) Need information about internal structure of graph

38. Circular planar graphs 1 1 4 1 3 4 3 5 5 4 2 2 3 planar, not circular planar 2 circular planar circular planar Planar graph Special vertices called nodes on outer face Nodes numbered in counterclockwise order along outer face

41. 3 2 1 2 4 3 1 4 4 3 2 1