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- W2963868968 abstract "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 [26] 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." @default.
- W2963868968 created "2019-07-30" @default.
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- W2963868968 date "2016-07-01" @default.
- W2963868968 modified "2023-10-14" @default.
- W2963868968 title "Random $mathbb{Z}^d$-shifts of finite type" @default.
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- W2963868968 doi "https://doi.org/10.3934/jmd.2016.10.287" @default.
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