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 )

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

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

Carroll-Speyer groves

Goal: compute the probability distribution of partition from random grove

Noncrossing (planar) partitions

Uniformly random grove

Multichordal loop-erased random walk

Peano curves surrounding trees

Double-dimer configuration

Noncrossing (planar) pairings

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

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

Grove partition probabilities

Double-dimer pairing probabilities

Planar partitions & planar pairings

Bilinear form on planar partitions / planar pairings

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

Bilinear form on planar partitions / planar pairings

These equivalences are enough to compute any column!

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

Grove partition probabilities

Dual electric network & dual partition Planar graph Dual graph Grove Dual grove

Curtis-Ingerman-Morrow formula Fomin gives another version of this formula, with combinatorial proof

Pfaffian formula

Caroll-Speyer groves

Assume nodes alternate black/white

arXiv:math.PR/