Random trees with superexponential branching weights

Details Random trees with superexponential branching weights Random trees with superexponential branching weights

2011-04-14

arxiv.org/abs/1104.2810v1

J. Phys. A: Math. Theor. 44 (2011) 485002

Physics

Mathematical Physics

19 pages

Scientific paper

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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