Download presentation
Presentation is loading. Please wait.
Published byTerence Alexander Modified over 9 years ago
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
22
Double-dimer pairing probabilities
24
Planar partitions & planar pairings
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
31
These equivalences are enough to compute any column!
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
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
41
Assume nodes alternate black/white
45
arXiv:math.PR/0608422
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.