Spanning Forests on Random Planar Lattices

TL;DR

This paper develops a random-matrix approach to spanning forests on random planar lattices, revealing connections with the Potts and O(n) models, and provides exact resummation techniques for the expansion in the number of forest components.

Contribution

It introduces a novel random-matrix formulation for spanning forests on random planar lattices and establishes links with the Potts and O(n) models, enabling exact resummation methods.

Findings

Connection between spanning forests and Potts/O(n) models.

Exact resummation of the expansion in forest components.

Identification of a special class of terms with exact resummation.

Abstract

The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q -> 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n -> -1, the expansion parameter t counting the number of components in the forest. We give a random-matrix formulation of this model on the ensemble of degree-k random planar lattices. For k = 3, a correspondence is found with the Kostov solution of the loop-gas problem, which arise as a reformulation of the (logarithmic action) O(n) model, at n = -2. Then, we show how to perform an expansion around the t = 0 theory. In the thermodynamic limit, at any order in t we have a finite sum of finite-dimensional Cauchy integrals. The leading contribution comes from a peculiar class of terms, for which a resummation can be performed exactly.