Random $\mathbb{Z}^d$-shifts of finite type

In this work we consider an ensemble of random $\mathbb{Z}^d$-shifts of finite type ($\mathbb{Z}^d$-SFTs) and prove several results concerning the behavior of typical systems with respect to emptiness, entropy, and periodic points. These results generalize statements made in \cite{McGoff} regarding the case $d=1$. Let $\mathcal{A}$ be a finite set, and let $d \geq 1$. For $n$ in $\mathbb{N}$ and $α$ in $[0,1]$, define a random subset $ω$ of $\mathcal{A}^{[1,n]^d}$ by independently including each pattern in $\mathcal{A}^{[1,n]^d}$ with probability $α$. Let $X_ω$ be the (random) $\mathbb{Z}^d$-SFT built from the set $ω$. For each $α\in [0,1]$ and $n$ tending to infinity, we compute the limit of the probability that $X_ω$ is empty, as well as the limiting distribution of entropy of $X_ω$. Furthermore, we show that the probability of obtaining a nonempty system without periodic points tends to zero. For $d>1$, the class of $\mathbb{Z}^d$-SFTs is known to contain strikingly different behavior than is possible within the class of $\mathbb{Z}$-SFTs. Nonetheless, the results of this work suggest a new heuristic: typical $\mathbb{Z}^d$-SFTs have similar properties to their $\mathbb{Z}$-SFT counterparts.

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