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 w n of a vertex of degree n is asymptotic to n −β .
We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n −1/2 is described by a Brownian excursion. The planar maps, with the graph metric rescaled by n −1/2 , are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.
This paper concerns random bipartite planar maps which are defined by assigning weights to their faces. The paper presents a threefold contribution to the theory. Firstly, we prove the existence of the local limit for all choices of weights and describe it in terms of an infinite mobile. Secondly, we show that the local limit is in all cases almost surely recurrent. And thirdly, we show that for certain choices of weights the local limit has exactly one face of infinite degree and has in that case spectral dimension $4/3$ (the latter requires a mild moment condition).Comment: 47 pages, 6 figures. Revised versio
We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model generalises the preferential attachment model and Ford's α-model for phylogenetic trees. We develop a mean field theory for the vertex degree distribution, prove that the mean field theory is exact in some special cases and check that it agrees with numerical simulations in general. We calculate various correlation functions and show that the intrinsic Hausdorff dimension can vary from one to infinity, depending on the parameters of the model.
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