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Boundary Partitions in Trees and Dimers

Boundary Partitions in Trees and Dimers. (Connection probabilities in multichordal SLE 2 , SLE 4 , and SLE 8 ). Richard W. Kenyon and David B. Wilson. University of British Columbia. Microsoft Research. Multichordal SLE. Crossing probabilities:. Percolation -- Cardy ’92

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Boundary Partitions in Trees and Dimers

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  1. Boundary Partitions in Trees and Dimers (Connection probabilities in multichordal SLE2, SLE4, and SLE8) Richard W. Kenyon and David B. Wilson University of British Columbia Microsoft Research

  2. Multichordal SLE Crossing probabilities: Percolation -- Cardy ’92 Smirnov ’01 Critical Ising – Arguin & Saint-Aubin ’02 Bichordal SLE -- Bauer, Bernard, Kytölä ’05 Trichordal SLE6, multichordal SLE – Dubédat ’05 Covariant measure for parallel crossing -- Kozdron & Lawler ’06 Multichordal SLE2, SLE4, SLE8, double-dimer paths – Kenyon & W ’06 SLE4 characterization of discrete Guassian free field – Schramm & Sheffield ’06

  3. 1 3 5 4 2 1 3 5 4 2 1 3 5 4 2 Spanning forest rooted at {1,2,3} Spanning tree Planar graph Special vertices called nodes on outer face Nodes numbered in counterclockwise order along outer face Kirchoff matrix (negative Laplacian) Matrix-tree theorem

  4. 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

  5. Carroll-Speyer groves

  6. 1 3 5 4 2 Goal: compute the probability distribution of partition from random grove

  7. Noncrossing (planar) partitions 4 4 1 3 1 3 2 2 4 1 3 2

  8. Uniformly random grove

  9. Multichordal loop-erased random walk

  10. Peano curves surrounding trees

  11. Double-dimer configuration

  12. Noncrossing (planar) pairings 4 4 1 3 1 3 2 2 4 1 3 2

  13. Double-dimer model in upper half plane with nodes at integers

  14. (negative of) Dirichlet-to-Neumann matrix Electric network

  15. 1 3 5 4 2

  16. 1 3 5 4 2 0

  17. 4 1 3 2

  18. 4 1 3 2

  19. Grove partition probabilities

  20. Double-dimer pairing probabilities

  21. Planar partitions & planar pairings

  22. Planar partitions & planar pairings

  23. Bilinear form onplanar partitions / planar pairings

  24. Ko & Smolinsky determine when matrix is singular Gram Matrix of Temperley-Lieb Algebra Meander Matrix Di Francesco, Golinelli, Guitter diagonalize matrix

  25. Bilinear form onplanar partitions / planar pairings

  26. These equivalences are enough to compute any column!

  27. Computing column  By induction find equivalent linear combination when item n deleted from . If {n} is a part of , use rule for adjoining new part. Otherwise, n is in same part as some other item j, use splitting rule. n n Now induct on # parts that cross part containing j & n Use crossing rule with part closest to j j

  28. Grove partition probabilities

  29. 3 1 3 5 4 3 1 2 4 2 1 2 4 1 3 5 4 2 Dual electric network & dual partition Planar graph Dual graph Grove Dual grove

  30. Curtis-Ingerman-Morrow formula 1 8 2 7 3 6 4 5 Fomin gives another version of this formula, with combinatorial proof

  31. Pfaffian formula 6 5 1 4 2 3

  32. Caroll-Speyer groves

  33. Caroll-Speyer groves

  34. Assume nodes alternate black/white

  35. arXiv:math.PR/0608422

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