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A large-deviation theorem for tree-indexed Markov chains

Amir Dembo·Peter Morters·Scott Sheffield
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Abstract

Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed by taking all descendants of a single vertex. We prove a large deviation principle in $n$, with explicit rate function, for the empirical subtree measures of multitype Galton-Watson trees conditioned to have exactly $n$ vertices. In the process, we extend the notions of shift-invariance and specific relative entropy--as typically understood for Markov fields on deterministic graphs such as $\mathbb Z^d$--to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with $n$ vertices.

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Amir Dembo, Peter Morters, Scott Sheffield. 2003-06-02. A large-deviation theorem for tree-indexed Markov chains. https://arxiv.org/abs/math/0306045

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