boundary partitions in trees and dimers richard w. kenyon and david b. wilson university of british...

Boundary Partitions in Trees and Dimers

Richard W. Kenyon and David B. Wilson

University of British Columbia Brown University

Microsoft Research

(Connection probabilities in multichordal SLE2, SLE4, and SLE8)

arXiv:math.PR/0608422

Boundary connections(Razumov & Stroganov)

Exponents from networks(Duplantier & Saleur)

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

Arbitrary finite graph with two special nodes

Kirchoff’s formula for resistance

3 spanning trees

5 2-tree forests with nodes 1 and 2 separated

5 4

2

1 3

5 4

2

1 3

Spanning tree

Kirchoff matrix (negative Laplacian)

5 4

2

1 3

Spanning forestrooted at {1,2,3}

Matrix-tree theorem (Kirchoff)

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3

5 4

2

1 3Arbitrary finite graph with two special nodes

(Kirchoff)

3

three

Arbitrary finite graph with four special nodes?

5

32

1 4 All pairwise resistances are equal

32

1 4 All pairwise resistances are equal

Need more than boundary measurements (pairwise resistances)Need information about internal structure of graph

5 4

2

1 3

Planar graphSpecial vertices called nodes on outer faceNodes numbered in counterclockwise order along outer face

Circular planar graphs

5

32

1 4

circular planarcircular planar

3

2

1

4

planar,not circular planar

Noncrossing (planar) partitions

2

1 3

4

2

1 3

4

2

1 3

4

5 4

2

1 3

Goal: compute the probability distribution of partition from random grove

Carroll-Speyer groves

Carroll-Speyer ’04

Petersen-Speyer ’05

Multichordal SLE

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

Crossing probabilities:

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 ’06Surprising connection between =4 and =8,2

Uniformly random grove

Peano curves surrounding trees

Multichordal loop-erased random walk

Double-dimer configuration

Noncrossing (planar) pairings

2

1 3

4

2

1 3

4

2

1 3

4

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

Contours in discrete Gaussian free field(Schramm & Sheffield)

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)

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

5 4

2

1 3

5 4

2

1 3

0

1

2

4

3

1

2

4

3

Grove partition probabilities

Bilinear form onplanar partitions / planar pairings

Meander MatrixGram Matrix of Temperley-Lieb AlgebraKo & Smolinsky determine when matrix is singularDi Francesco, Golinelli, Guitter diagonalize matrix

Bilinear form onplanar partitions / planar pairings

These equivalences are enough to compute any column!

(extra term in recent work by Caraciollo-Sokal-Sportiello on hyperforests)

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

nnNow induct on # parts thatcross part containing j & n

Use crossing rule withpart closest to j

Grove partition probabilities

Dual electric network & dual partition

5 4

2

1 3

1 2

3

4

Planar graph Dual graph

Grove Dual grove

1 2

3

4

5 4

2

1 3

Curtis-Ingerman-Morrow formula

1

2

3

4

8

7

6

5

Fomin gives another version of this formula, with combinatorial proof

Pfaffian formula

1

2 3

4

56

Double-dimer pairing probabilities

Planar partitions & planar pairings

Planar partitions & planar pairings

Assume nodes alternate black/white

arXiv:math.PR/0608422

Caroll-Speyer groves

Caroll-Speyer groves