Typicality and entropy of processes on infinite trees

Backhausz, Ágnes and Bordenave, C. and Szegedy, Balázs (2022) Typicality and entropy of processes on infinite trees. ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 58 (4). pp. 1959-1980. ISSN 0246-0203

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Abstract

Consider a uniformly sampled random d-regular graph on n vertices. If d is fixed and n goes to ∞ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called "typical"processes) on the infinite d-regular tree Td. This correspondence between ergodic theory on Td and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processes on Td. It serves as a quantitative refinement of the notion of typicality and is tightly connected to the asymptotic free energy in statistical physics. Using entropy inequalities, we provide new sufficient conditions for typicality for edge Markov processes. We also extend these notions and results to processes on unimodular Galton. Watson random trees. © 2022 Association des Publications de l'Institut Henri Poincaré.

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