Abstract
Fix $n\in\mathbb{N}$. Let $\mathbf{T}_{n}$ be the set of rooted trees $(T,o)$ whose vertices are labeled by elements of $\{1,\dots,n\}$. Let $\nu$ be a strongly connected multi-type Galton–Watson measure. We give necessary and sufficient conditions for the existence of a measure $\mu$ that is reversible for simple random walk on $\mathbf{T}_{n}$ and has the property that given the labels of the root and its neighbors, the descendant subtrees rooted at the neighbors of the root are independent multi-type Galton–Watson trees with conditional offspring distributions that are the same as the conditional offspring distributions of $\nu$ when the types are $\nu$ are ordered pairs of elements of $[n]$. If the types of $\nu$ are given by the labels of vertices, then we give an explicit description of such $\mu$.
Citation
Serdar Altok. "Unimodularity for multi-type Galton–Watson trees." Bernoulli 19 (3) 780 - 802, August 2013. https://doi.org/10.3150/11-BEJ416
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