Condensation in nongeneric trees

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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 $n^{-\beta}$.