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 $\alpha$ in $[0,1]$, define a random subset $\omega$ of $\mathcal{ A}^{ [1,n]^d}$ by independently including each pattern in $\mathcal{ A}^{ [1,n]^d}$ with probability $\alpha$. Let $X_{ \omega}$ be the (random) $\mathbb{ Z}^d$-SFT built from the set $\omega$. For each $\alpha \in [0,1]$ and $n$ tending to infinity, we compute the limit of the probability that $X_{ \omega}$ is empty, as well as the limiting distribution of entropy of $X_{ \omega}$. 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.
Categories: math.DS math.PR