Ramanujan subshifts

TL;DR

This paper generalizes Ramanujan graphs to higher-dimensional subshifts, constructing explicit examples with optimal correlation decay and producing infinite families of Ramanujan graphs via automata.

Contribution

It introduces the concept of Ramanujan subshifts in higher dimensions and constructs explicit examples using quaternionic lattices, linking subshifts to Ramanujan graphs.

Findings

Existence of q-regular Ramanujan Z^δ-subshifts for odd prime powers q

Construction of explicit Ramanujan graphs from these subshifts

Development of a local automaton-based neighbor computation method

Abstract

A finite, connected, $(d+1)$-regular graph $G$ is called Ramanujan if every its eigenvalue $\lambda$ satisfies either $\lambda=\pm (d+1)$ or $|\lambda|\leq 2\sqrt{d}$. 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 $\mathbb{Z}^{\delta}$-subshift of finite type, and we define a Ramanujan subshift as a $d$-regular $\mathbb{Z}^{\delta}$-subshift with an optimal rate of decay of correlations. We show that for every odd prime power $q\geq 3$ and dimension $\delta<q$, there exists a $q$-regular Ramanujan $\mathbb{Z}^{\delta}$-subshift. The construction is based on the quaternionic lattices over $\mathbb{F}_q(t)$ introduced by Rungtanapirom-Stix-Vdovina (2019). Each of our $q$-regular Ramanujan subshifts gives rise to a family of non-bipartite $(q+1)$-regular Ramanujan graphs. These graphs are very explicit and local in the strong sense: the neighbors of any vertex can be computed by an explicit Mealy automaton associated with the subshift. As a byproduct, for every odd prime power $q$, we get a single lifting rule that can be iterated to produce an infinite family of $(q+1)$-regular Ramanujan graphs.