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arXiv · 1104.2810

Random trees with superexponential branching weights

Svante Janson·Thordur Jonsson·Sigurdur Orn Stefansson
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Abstract

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case when $w_n$ grows faster than exponentially with $n$. In this case the measures on trees of finite size $N$ converge weakly as $N$ tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form $w_n=((n-1)!)^\alpha$ with $\alpha >0$ we obtain more refined results about the approach to the infinite volume limit.

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Svante Janson, Thordur Jonsson, Sigurdur Orn Stefansson. 2011-04-14. Random trees with superexponential branching weights. https://doi.org/10.1088/1751-8113/44/48/485002

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