Random subshifts of finite type

Abstract

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 ≤ α ≤ 1 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.