Random subshifts of finite type

TL;DR

This paper investigates the probabilistic properties of random subshifts of finite type, analyzing their likelihood of being empty and their entropy distribution as the word length grows, revealing phase transitions and structural behaviors.

Contribution

It introduces a new probabilistic model for subshifts of finite type and derives asymptotic results on their emptiness and entropy distribution, differing from previous models.

Findings

Probability of emptiness converges as n increases.

Likelihood of a unique positive entropy component approaches 1 exponentially.

Entropy distribution of the random SFTs is characterized asymptotically.

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 $\omega$ of $B_n(X)$ by independently choosing each word from $B_n(X)$ with some probability $\alpha$. Let $X_{\omega}$ be the (random) SFT built from the set $\omega$. For each $0\leq \alpha \leq1$ and $n$ tending to infinity, we compute the limit of the likelihood that $X_{\omega}$ is empty, as well as the limiting distribution of entropy for $X_{\omega}$. For $\alpha$ near 1 and $n$ tending to infinity, we show that the likelihood that $X_{\omega}$ 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.