Pressure and escape rates for random subshifts of finite type

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.