François Baccelli, Indistinguishability of Connected Components and Generations in Directed Forests скачать в хорошем качестве

François Baccelli, Indistinguishability of Connected Components and Generations in Directed Forests

This talk is focused on indistinguishability properties of classes w.r.t. the two most basic equivalence relations on infinite random directed unimodular forests, namely the “connectivity” relation, with equivalence classes the component trees of the forest, and the “same level” relation, with equivalence classes the generations in each component. The main focus of the paper is on forests in which all component trees are one-ended, which is known to be a fundamental subclass among two in this context. This includes the class of random undirected forests where all component trees are one-ended, which is ubiquitous. The components (resp. generations) are said to be “distinguishable” if one can find a non-trivial property of the components (resp. generations) satisfied by certain components (resp. generations) and not by others. The main results of the paper are the indistinguishability of the components and that of the generations for a class of “random enough” unimodular forests. This contains the class of “coalescing Markov trajectories on unimodular random graphs”, defined in the paper, where the directed edges defining the forest are conditionally independent given the underlying graph. This also contains models where the edges are deterministic given the graph, but the underlying unimodular random graph enjoys local randomness and independence properties, as, for instance, the support of a Poisson point process of the Euclidean space or a Bernoulli point process on a lattice. This leads to new indistinguishability results for these equivalence classes in a variety of models discussed in the literature, including, for instance, river models, coalescing renewal process models, as well as for models based on point maps on Poisson point processes. It also leads to a new and simpler proof of the indistinguishability of the components of the wired uniform spanning forest. The proof technique, which is original, is based on a conditioning on the “ancestry chain” of the root. It leverages measure-theoretic results on the completion of certain invariant σ-algebras, new general results on directed unimodular forests, as well as new results on Markov chains on unimodular random graphs, which are all of independent interest. The indistinguishability of the components is proved by establishing the triviality of the invariant σ-algebra of the ancestry chain of the root, whereas that of the generations follows from the triviality of the tail σ-algebra of this chain. These triviality properties hold true under general conditions which are identified and which are shown to be satisfied by the examples listed above.