Condensation in nongeneric trees

Details Condensation in nongeneric trees Condensation in nongeneric trees

2010-09-09

arxiv.org/abs/1009.1826v2

Physics

Condensed Matter

Statistical Mechanics

57 pages, 14 figures. Minor changes

Scientific paper

10.1007/s10955-010-0104-8

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 $2\beta -2$ if the weight $w_n$ of a vertex of degree $n$ is asymptotic to $n^{-\beta}$.

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