Random ℤd-shifts of finite type

Abstract

In this work we consider an ensemble of random ℤd -shifts of finite type (ℤd -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 A be a finite set, and let d ≥ 1. For n in ℕ and α in [0,1], define a random subset ω of A[1,n]d by independently including each pattern in A[1,n]d with probability α. Let Xω be the (random) ℤd -SFT built from the set ω. For each α ε [0,1] and n tending to infinity, we compute the limit of the probability that Xω is empty, as well as the limiting distribution of entropy of Xω. Furthermore, we show that the probability of obtaining a nonempty system without periodic points tends to zero. For d > 1, the class of ℤd -SFTs is known to contain strikingly different behavior than is possible within the class of ℤ-SFTs. Nonetheless, the results of this work suggest a new heuristic: typical ℤd -SFTs have similar properties to their ℤ-SFT counterparts.