Spanning Forests on Random Planar Lattices

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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 of 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.

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