Transcript rsws1 6736

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
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
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Spanning forest
Spanning tree
rooted at {1,2,3}
Planar graph
Special vertices called nodes on outer face
Nodes numbered in counterclockwise order along outer face
Kirchoff matrix (negative Laplacian)
Matrix-tree theorem
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Carroll-Speyer groves
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Goal: compute the probability distribution of
partition from random grove
Noncrossing (planar) partitions
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Uniformly random grove
Multichordal loop-erased random walk
Peano curves surrounding trees
Double-dimer configuration
Noncrossing (planar) pairings
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Double-dimer model in upper half
plane with nodes at integers
(negative of)Electric
Dirichlet-to-Neumann
matrix
network
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Grove partition probabilities
Double-dimer pairing probabilities
Planar partitions & planar pairings
Planar partitions & planar pairings
Bilinear form on
planar partitions / planar pairings
Ko & Smolinsky
determine
when matrix isAlgebra
singular
Gram
Matrix
Meander
of
Temperley-Lieb
Matrix
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.
n
Now induct on # parts that
cross part containing j & n
Use crossing rule with
part closest to j
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Grove partition probabilities
Dual electric network & dual partition
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Planar graph
Dual graph
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Grove
Dual grove
Curtis-Ingerman-Morrow formula
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Fomin gives another version of this formula, with combinatorial proof
Pfaffian formula
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Caroll-Speyer groves
Caroll-Speyer groves
Assume nodes alternate black/white
arXiv:math.PR/0608422