Dimension gap and phase transition for one-dimensional random walks with reflective boundary

TL;DR

This paper investigates the dimension gap and phase transitions in one-dimensional random walks with reflective boundaries, providing variational formulas and conditions for phase transitions based on Gurevich pressure analysis.

Contribution

It introduces new variational formulas for Gurevich pressure for systems with and without reflective boundaries, and characterizes conditions for phase transitions and dimension gaps.

Findings

Characterization of systems with a dimension gap.

Conditions for second order phase transitions with reflective boundaries.

Derivation of a variational formula for the spectral radius of infinite Hessenberg matrices.

Abstract

We study $\mathbb Z$- and $\mathbb N$-extensions of interval maps with at most countably many full branches modelling one-dimensional random walks without and with a reflective boundary. We analyse the associated Gurevich pressure and explore the relations governing these two cases. For such extensions, we obtain variational formulae for the Gurevich pressure that depend only on the base system. As a consequence, we characterise the systems with a dimension gap and, in the presence of a reflective boundary, provide general conditions in terms of asymptotic covariances for a second order phase transition. As a by-product, we derive a variational formula for the spectral radius of infinite Hessenberg matrices.