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Boundary Partitions in Trees and Dimers Richard W. Kenyon and David B. Wilson University of British ColumbiaMicrosoft Research (Connection probabilities.

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Presentation on theme: "Boundary Partitions in Trees and Dimers Richard W. Kenyon and David B. Wilson University of British ColumbiaMicrosoft Research (Connection probabilities."— Presentation transcript:

1 Boundary Partitions in Trees and Dimers Richard W. Kenyon and David B. Wilson University of British ColumbiaMicrosoft Research (Connection probabilities in multichordal SLE 2, SLE 4, and SLE 8 )

2 Multichordal SLE Percolation -- Cardy ’92 Smirnov ’01 Critical Ising – Arguin & Saint-Aubin ’02 Bichordal SLE  -- Bauer, Bernard, Kytölä ’05 Trichordal SLE 6, multichordal SLE  – Dubédat ’05 Covariant measure for parallel crossing -- Kozdron & Lawler ’06 Crossing probabilities: Multichordal SLE 2, SLE 4, SLE 8, double-dimer paths – Kenyon & W ’06 SLE 4 characterization of discrete Guassian free field – Schramm & Sheffield ’06

3 54 2 13 Planar graph Special vertices called nodes on outer face Nodes numbered in counterclockwise order along outer face 54 2 13 Spanning tree Kirchoff matrix (negative Laplacian) Matrix-tree theorem 54 2 13 Spanning forest rooted at {1,2,3}

4 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13

5 Carroll-Speyer groves

6 54 2 13 Goal: compute the probability distribution of partition from random grove

7 Noncrossing (planar) partitions 2 13 4 2 13 4 2 13 4

8 Uniformly random grove

9 Multichordal loop-erased random walk

10 Peano curves surrounding trees

11 Double-dimer configuration

12 Noncrossing (planar) pairings 2 13 4 2 13 4 2 13 4

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

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

15 54 2 13

16 54 2 13 0

17 1 2 4 3

18 1 2 4 3

19 Grove partition probabilities

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22 Double-dimer pairing probabilities

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24 Planar partitions & planar pairings

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26 Bilinear form on planar partitions / planar pairings

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

28 Bilinear form on planar partitions / planar pairings

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31 These equivalences are enough to compute any column!

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33 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. j n n Now induct on # parts that cross part containing j & n Use crossing rule with part closest to j

34 Grove partition probabilities

35 Dual electric network & dual partition 54 2 13 1 2 3 4 Planar graph Dual graph Grove Dual grove 1 2 3 4 54 2 13

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

38 Pfaffian formula 1 2 3 4 56

39 Caroll-Speyer groves

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41 Assume nodes alternate black/white

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45 arXiv:math.PR/0608422


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