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Local Statistics of Uniform Spanning Trees and Sandpile Models

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winka
41 diapositives · 552 Views · Jul 26, 2014

This study explores the intricate relationship between uniform spanning trees (UST) and sandpile models within the context of probability theory and graphical analysis. It presents bijections and fundamental properties of spanning trees, along with computations of topologically defined events and their relevance to UST. The paper discusses local statistics and conjectures regarding sandpile densities on various lattices, using methodologies like resistance and electrical quantities. Key probabilities are evaluated, providing insights into the behaviors of these mathematical structures and their connections.

#local statistics of uniform spanning trees and sandpile models #local statistics of uniform spanning trees and sandpile models ppt #powerpoint #presentation
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  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]

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

  5. Priezzhev (’94)

  6. Jeng—Piroux—Ruelle (’06)

  7. [burning bijection demo]

  8. Underlying graph Uniform spanning tree

  9. 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

  10. 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

  11. 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?

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

  13. 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)

  14. Kenyon—W

  15. Kenyon—W

  16. Kenyon—W

  17. Kenyon—W

  18. Joint distribution of heights at two neighboring vertices

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

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

  21. Sandpiles on triangular lattice

  22. 1 2 4 3

  23. 2 3 4 1

  24. 3 4 1 2

  25. 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

  26. Uniformly random grove

  27. 1 3 5 4 2 Goal: compute ratios of partition functions in terms of electrical quantities

  28. 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

  29. 1 3 3 Arbitrary finite graph with two special nodes three 5 4 (Kirchoff) 2

  30. 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

  31. 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

  32. 3 2 1 2 4 3 1 4 4 3 2 1