Unimodularity for multi-type Galton-Watson trees

摘要

Fix n σ N. Let Tn be the set of rooted trees (T,o) whose vertices are labeled by elements of {1, ..., n}. Let ? be a strongly connected multi-type Galton-Watson measure. We give necessary and sufficient conditions for the existence of a measure μ that is reversible for simple random walk on Tn 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 ? when the types are ? are ordered pairs of elements of [n]. If the types of ? are given by the labels of vertices, then we give an explicit description of such μ.