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Random subshifts of finite type

Let $X$ be an irreducible shift of finite type (SFT) of positive entropy, and let $B_n(X)$ be its set of words of length $n$. Define a random subset $ω$ of $B_n(X)$ by independently choosing each word from $B_n(X)$ with some probability $α$. Let $X_ω$ be the (random) SFT built from the set $ω$. For each $0\leq α\leq1$ and $n$ tending to infinity, we compute the limit of the likelihood that $X_ω$ is empty, as well as the limiting distribution of entropy for $X_ω$. For $α$ near 1 and $n$ tending to infinity, we show that the likelihood that $X_ω$ contains a unique irreducible component of positive entropy converges exponentially to 1. These results are obtained by studying certain sequences of random directed graphs. This version of "random SFT" differs significantly from a previous notion by the same name, which has appeared in the context of random dynamical systems and bundled dynamical systems.