Ramanujan subshifts

A finite, connected, $(d+1)$-regular graph $G$ is called Ramanujan if every its eigenvalue $λ$ satisfies either $λ=\pm (d+1)$ or $|λ|\leq 2\sqrt{d}$. The Ramanujan condition corresponds to the optimal rate of decay of correlations for the associated non-backtracking edge subshift. We consider a higher-dimensional generalization of this observation. We introduce the notion of a $d$-regular $\mathbb{Z}^δ$-subshift of finite type, and we define a Ramanujan subshift as a $d$-regular $\mathbb{Z}^δ$-subshift with an optimal rate of decay of correlations. We show that for every odd prime power $q\geq 3$ and dimension $δ