2010/09/30 by Thordur Jonsson, Thórdur Jónsson, Sigurður Örn Stefánsson +1
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP #msc:60J80
paper · pdf · doi:10.1007/s10955-010-0104-8
57 pages, 14 figures. Minor changes
openalex publication_date 2010/12/09 · arxiv created 2011/01/04 · arxiv updated 2011/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability m<1. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like (1-m)N where N is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2β-2 if the weight wn of a vertex of degree n is asymptotic to n-β.