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