Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\alpha$ 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.

4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci