Spanning Forests on Random Planar Lattices

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.

43 pages, Dedicated to Edouard Brezin and Giorgio Parisi, on the occasion of their special birthday

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Cited by in corpus (6)

• Matrix integrals $\&$ finite holography
• Spanning forests in regular planar maps
• Phase transition in the spanning-hyperforest model on complete hypergraphs
• A Flow in the Forest
• Critical Behaviour of Spanning Forests on Random Planar Graphs
• Dualities between fermionic theories and the Potts model