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Cutting down $$\mathbf{p}$$-trees and inhomogeneous continuum random trees. (English) Zbl 1388.60036
This paper is about random rooted trees. We initially consider a finite vertex set $$A$$ and a probability distribution $$\mathbf{p}=(p_{1},p_{2},.\dots p_{| A|})$$ on $$A$$ (with all probabilities strictly positive) and select tree $$T$$ with probability $$\prod_{u\in A}p_{u}^{C_{u}(t)}$$ where $$C_{u}(t)$$ is the indegree of vertex $$u$$ in tree $$t$$ when we think of edges as being oriented towards the root. (This is a probability distribution, by a version of Cayley’s formula). These are the $$\mathbf{p}$$-trees of M. Camarri and J. Pitman [Electron. J. Probab. 5, Paper No. 1, 19 p. (2000; Zbl 0953.60030)].
The tree is cut down to the root by choosing a vertex at random, according to $$\mathbf{p}$$ conditioned on the remaining part containing a random vertex $$V$$ according to $$\mathbf{p}$$ at each stage, and we retain the component which contains $$V$$. Let $$L(T)$$ denote the time until $$V$$ is chosen. There is a tree which encodes this fragmentation process, and one theme of the paper is to demonstrate certain exact correspondences between the original tree and the encoding tree.
The limit objects of $$\mathbf{p}$$-trees are inhomogeneous continuum random trees (Aldous’ original Brownian continuum random tree corresponds to the case of $$\mathbf{p}$$ being uniform on $$A$$). The results referred to at the end of the last paragraph allow distributional correspondences between the initial inhomogeneous continuum random tree and a similar tree which encodes the fragmentation.

##### MSC:
 60C05 Combinatorial probability 05C80 Random graphs (graph-theoretic aspects) 60F17 Functional limit theorems; invariance principles
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