Spanning Forests on Random Planar Lattices

作者：Caracciolo Sergio^*; Sportiello Andrea

来源：Journal of Statistical Physics, 2009, 135(5-6): 1063-1104.

DOI：10.1007/s10955-009-9733-1

• Random matrices
• Potts model
• O(n)-invariant sigma-model
• O(n)-vector model
• Spanning trees
• Spanning forests
• Self-avoiding polymers

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.

• 出版日期2009-6