Recurrence on the average on trees

TL;DR

This paper proves that certain infinite trees with bounded coordination and negligible surface-to-volume ratio are recurrent on average, impacting the understanding of symmetry breaking in statistical models on these trees.

Contribution

It establishes a recurrence property for a class of infinite trees with bounded coordination and negligible surface, linking geometric properties to probabilistic behavior.

Findings

Infinite trees with bounded coordination are recurrent on the average.

Surface negligible compared to volume implies recurrence.

Results influence understanding of symmetry breaking in statistical models.

Abstract

In this paper we show that all infinite trees which have bounded coordination and whose surface is negligible with respect to the volume in the limit of large distances (so that they can be embedded in a finite-dimensional euclidean space) are recurrent on the average; this has important consequences about the spontaneous symmetry breaking of statistical models defined on such trees.