2009/03/25 by Sergio Caracciolo, Andrea Sportiello · 1 citation
Mathematics · Physics and Astronomy · #Cauchy distribution #Class (philosophy) #Equivalence (formal languages) #Generating function #Lattice (music) #Limit (mathematics) #Planar #Potts model #Random Matrices and Applications #Resummation #Spanning tree #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #hep-lat #hep-th #math-ph #math.CO #math.MP
paper · pdf · doi:10.1007/s10955-009-9733-1
published as J.Statist.Phys.135:1063-1104,2009 · 43 pages, Dedicated to Edouard Brezin and Giorgio Parisi, on the occasion of their special birthday
arxiv created 2009/03/25 · openalex publication_date 2009/04/07 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.