Boundary Partitions in Trees and Dimers

1
Boundary Partitions in Trees and Dimers
(Connection probabilities in multichordal SLE2,
SLE4, and SLE8)
arXivmath.PR/0608422

• Richard W. Kenyon and David B. Wilson

University of British Columbia Brown
University
Microsoft Research
2
Boundary connections(Razumov Stroganov)
3
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4
Exponents from networks(Duplantier Saleur)
5
Kirchoffs formula for resistance
1
3
Arbitrary finite graph with two special nodes
5
4
2
1
3
1
3
1
3
1
3
1
3
5
4
5
4
5
4
5
4
5
4
2
2
2
2
2
5 2-tree forests with nodes 1 and 2 separated
1
3
1
3
1
3
5
4
5
4
5
4
2
2
2
3 spanning trees
6
Matrix-tree theorem (Kirchoff)
1
3
5
4
2
Spanning tree
Spanning forest rooted at 1,2,3
Kirchoff matrix (negative Laplacian)
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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
8
1
3
3
Arbitrary finite graph with two special nodes
three
5
4
(Kirchoff)
2
9
Arbitrary finite graph with four special nodes?
1
4
All pairwise resistances are equal
5
3
2
1
4
All pairwise resistances are equal
3
2
Need more than boundary measurements (pairwise
resistances) Need information about internal
structure of graph
10
Circular planar graphs
1
1
4
1
3
4
3
5
5
4
2
3
2
planar, not circular planar
2
circular planar
circular planar
Planar graph Special vertices called nodes on
outer face Nodes numbered in counterclockwise
order along outer face
11
Noncrossing (planar) partitions
4
4
1
3
1
3
2
2
4
1
3
2
12
1
3
5
4
2
Goal compute the probability distribution of
partition from random grove
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Carroll-Speyer groves
Carroll-Speyer 04 Petersen-Speyer 05
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Multichordal SLE
Crossing probabilities
Percolation -- Cardy 92
Smirnov 01
Critical Ising Arguin Saint-Aubin 02
Smirnov 06
Bichordal SLE? Bauer, Bernard, Kytölä 05
Trichordal percolation, 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
SLE and ADE (from CFT) Cardy 06
Surprising connection between ?4 and ?8,2
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Uniformly random grove
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Peano curves surrounding trees
18
Multichordal loop-erased random walk
19
Double-dimer configuration
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Noncrossing (planar) pairings
4
4
1
3
1
3
2
2
4
1
3
2
21
Double-dimer model in upper half plane with nodes
at integers
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Contours in discrete Gaussian free field(Schramm
Sheffield)
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DGFF vs double-dimer model

• DGFF has SLE4 contours (Schramm-Sheffield)
• Double-dimer believed to have SLE4 contours, no
  proof
• Connection probabilities are the same in the
  scaling limit (Kenyon-W 06)

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Electric network
(negative of) Dirichlet-to-Neumann matrix
25
1
3
5
4
2
26
1
3
5
4
2
0
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Grove partition probabilities
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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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(extra term in recent work by Caraciollo-Sokal-Spo
rtiello on hyperforests)
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
43
Pfaffian formula
5
6
1
4
2
3
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Double-dimer pairing probabilities
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Planar partitions planar pairings
48
Planar partitions planar pairings
49
Assume nodes alternate black/white
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arXivmath.PR/0608422
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Caroll-Speyer groves
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Caroll-Speyer groves