Boundary Partitions in Trees and Dimers

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
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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
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
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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
Electric network
(negative of) Dirichlet-to-Neumann matrix
15
1
3
5
4
2
16
1
3
5
4
2
0
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Grove partition probabilities
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Double-dimer pairing probabilities
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Planar partitions planar pairings
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Planar partitions planar pairings
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Bilinear form onplanar partitions / planar
pairings
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Meander Matrix
Ko Smolinsky determine when matrix is singular
Gram Matrix of Temperley-Lieb Algebra
Di Francesco, Golinelli, Guitter diagonalize
matrix
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Bilinear form onplanar partitions / planar
pairings
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These equivalences are enough to compute any
column!
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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
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Grove partition probabilities
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Dual electric network dual partition
Planar graph
Dual graph
Grove
Dual grove
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Curtis-Ingerman-Morrow formula
1
8
2
7
3
6
4
5
Fomin gives another version of this formula, with
combinatorial proof
38
Pfaffian formula
5
6
1
4
2
3
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Caroll-Speyer groves
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Caroll-Speyer groves
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Assume nodes alternate black/white
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arXivmath.PR/0608422