Condensation in nongeneric trees

Abstract

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 $n^{-\beta }$.

57 pages, 14 figures. Minor changes

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Cited by in corpus (22)

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• Local limits of conditioned Galton-Watson trees I: the infinite spine case
• Recurrence of bipartite planar maps
• Random walk on random infinite looptrees
• An introduction to Galton-Watson trees and their local limits
• Random trees with superexponential branching weights
• Tree limits and limits of random trees
• Percolation on random triangulations and stable looptrees
• Spectral dimension of trees with a unique infinite spine
• Ageing of complex networks
• Conditioning Galton-Watson trees on large maximal out-degree
• Markov branching in the vertex splitting model
• Voter models on subcritical inhomogeneous random graphs
• Critical Multi-Type Galton-Watson Trees Conditioned to be Large
• Generic Ising Trees
• Spectral dimension in graph models of causal quantum gravity
• Limits of the boundary of random planar maps
• Local limits of conditioned Galton-Watson trees II: the condensation case
• Simply generated trees, conditioned Galton--Watson trees, random allocations and condensation
• Critical trees are neither too short nor too fat
• Brownian continuum random tree conditioned to be large
• Conditioning Bienaym{é}-Galton-Watson trees to have large sub-populations