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