Pressure and escape rates for random subshifts of finite type

TL;DR

This paper investigates how the pressure and escape rates behave in random subshifts of finite type, showing convergence to specific values as the size of forbidden word sets grows and the probability parameter approaches one.

Contribution

It introduces a probabilistic framework for analyzing pressure and escape rates in random subshifts, revealing their asymptotic behavior as the system size increases.

Findings

Pressure converges to P_X(f) + log(α) in probability.

Escape rate converges to -log(α) in probability.

Results hold for α sufficiently close to 1 as n increases.

Abstract

In this work we consider several aspects of the thermodynamic formalism in a randomized setting. Let $X$ be a non-trivial mixing shift of finite type, and let $f : X \to \mathbb{R}$ be a H\"older continuous potential with associated Gibbs measure $\mu$. Further, fix a parameter $\alpha \in (0,1)$. For each $n \geq 1$, let $\mathcal{F}_n$ be a random subset of words of length $n$, where each word of length $n$ that appears in $X$ is included in $\mathcal{F}_n$ with probability $1-\alpha$ (and excluded with probability $\alpha$), independently of all other words. Then let $Y_n = Y(\mathcal{F}_n)$ be the random subshift of finite type obtained by forbidding the words in $\mathcal{F}_n$ from $X$. In our first main result, for $\alpha$ sufficiently close to $1$ and $n$ tending to infinity, we show that the pressure of $f$ on $Y_n$ converges in probability to the value $P_X(f) + \log(\alpha)$, where $P_X(f)$ is the pressure of $f$ on $X$. Additionally, let $H_n = H(\mathcal{F}_n)$ be the random hole in $X$ consisting of the union of the cylinder sets of the words in $\mathcal{F}_n$. For our second main result, for $\alpha$ sufficiently close to one and $n$ tending to infinity, we show that the escape rate of $\mu$-mass through $H_n$ converges in probability to the value $-\log(\alpha)$ as $n$ tends to infinity.