Finite orbits in random subshifts of finite type

TL;DR

This paper investigates the properties of random subshifts of finite type, showing that for small enough inclusion probability, these subshifts almost surely contain only finitely many elements as the size grows, especially in the one-dimensional case.

Contribution

It extends previous results by establishing a threshold probability for finiteness in random subshifts, including the optimal threshold in one dimension.

Findings

For $ ext{d} eq 1$, there exists $ ext{α}_0$ such that subshifts are finite with high probability.

In the case $ ext{d} = 1$, the threshold $ ext{α}_0 = 1/| ext{A}|$ is optimal.

The probability of finiteness approaches 1 as the size parameter $n$ tends to infinity.

Abstract

For each $n, d \in \mathbb{N}$ and $0 < \alpha < 1$, we define a random subset of $\mathcal{A}^{\{1, 2, \dots, n\}^d}$ by independently including each element with probability $\alpha$ and excluding it with probability $1-\alpha$, and consider the associated random subshift of finite type. Extending results of McGoff and of McGoff and Pavlov, we prove there exists $\alpha_0 = \alpha(d, |\mathcal{A}|) > 0$ such that for $\alpha < \alpha_0$ and with probability tending to $1$ as $n \to \infty$, this random subshift will contain only finitely many elements. In the case $d = 1$, we obtain the best possible such $\alpha_0$, $1/|\mathcal{A}|$.