Boundary partitions in trees and dimers

Abstract

Abstract. Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure; i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural algebraic interpretation and can be computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and, as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have a direct application to connection probabilities for multiple-strand SLE 2 , SLE 8 , and SLE 4 .

Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

“…2. In order to compute pairing probabilities of so-called the double-dimer model, Kenyon and Wilson [9,10] introduced a matrix M defined as follows. The rows and columns of M are indexed by λ, μ ∈ Dyck(2n), and M λ,μ = 1 if λ μ, and M λ,μ = 0 otherwise.…”

Determinantal spanning forests on planar graphs

“…by our condition on arg z, arg w. So we can remove the absolute values and conclude that the sum of the terms D p,q (2 − z i w j − z −i w −j ) k with (p, q) − (p 0 , q 0 ) < εn dominates the sum (15).…”

“…with (p, q) − (p 0 , q 0 ) < εn has larger exponential growth rate than any of the other terms in (15). So for n sufficiently large they dominate the sum (15). Moreover these terms have all approximately the same argument:…”

“…Choose z, w so that |z| = e x , |w| = e y and z i 0 w j 0 < 0 where both z and w have small arguments arg(z), arg(w) = O(1/n). Now consider the expressions (13) and (15) for the graph G n rather than G 1 . We claim that as n → ∞, Z concentrates on ERSFs of slope s, t in the following sense.…”

Generalized $$q$$ q -deformed correlation functions as spectral functions of hyperbolic geometry

“…This occurs in many areas of statistical physics [1][2][3] and topological string theory [4][5][6], BPS black holes, models of branes wrapping collapsed cycles in Calabi-Yau orbifolds, and quiver gauge theories [7][8][9]. Generalized MacMahon functions are used, in particular, in the computation of amplitudes of the A-model topological string [10][11][12][13], more specifically as regards the so-called topological vertex.…”