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 $δ<q$, there exists a $q$-regular Ramanujan $\mathbb{Z}^δ$-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.

💡 Research Summary

This paper presents a profound generalization of the well-known concept of Ramanujan graphs into the realm of higher-dimensional symbolic dynamics. A Ramanujan graph is a finite, connected, (d+1)-regular graph whose non-trivial adjacency eigenvalues are bounded in absolute value by 2√d. The authors begin by recalling a dynamical characterization: a graph is non-bipartite Ramanujan if and only if its associated non-backtracking edge subshift (a one-dimensional shift of finite type) exhibits an optimal exponential decay of correlations at the rate (1/√d)^n.

The core contribution of the paper is the introduction and construction of “Ramanujan subshifts” in higher dimensions. The authors first define a d-regular Z^δ-subshift of finite type as a δ-dimensional shift space where any rectangular pattern admits exactly d possible extensions in each coordinate direction. Such a subshift carries a natural uniform invariant measure. A Ramanujan subshift is then defined as a d-regular subshift that is strongly mixing and whose correlations decay at the optimal rate of (1/√d)^(∥n∥∞) as the lattice vector n in Z^δ grows, potentially with a polynomial prefactor.

The main existence theorem (Theorem 1) states that for every odd prime power q ≥ 3 and every dimension δ < q, there exists a q-regular Ramanujan Z^δ-subshift. The construction is explicit and arithmetic in nature, relying on the quaternionic lattices over the function field F_q(t) introduced by Rungtanapirom-Stix-Vdovina (2019). These lattices act simply transitively on the product of δ copies of the (q+1)-regular tree. The Ramanujan property of the resulting subshift is ultimately derived from the Ramanujan-Peterson conjecture for GL_2 over function fields, proved by Drinfeld.

A significant applied consequence of this dynamical construction is the generation of new families of expander graphs. Each q-regular Ramanujan Z^δ-subshift is linked to an explicit Mealy automaton (a finite-state transducer). The action graphs of this automaton, when restricted to reduced words of length n, yield non-bipartite (q+1)-regular Ramanujan graphs for all n ≥ 1 (Theorem 2). These graphs are “very explicit” and “local,” meaning the neighbors of any vertex can be computed efficiently by the fixed local rules of the automaton. Consequently, for each odd prime power q, the paper provides a single, deterministic lifting rule that can be iterated to produce an infinite family of (q+1)-regular Ramanujan graphs, offering a constructive counterpart to earlier non-constructive existence proofs.

The paper concludes by posing two central open problems: (1) Does a d-regular Ramanujan Z^δ-subshift exist for every d ≥ 2 and δ ≥ 2? (2) Does there exist a Mealy automaton (or a single lifting rule) generating d-regular Ramanujan graphs for every d ≥ 3? This work elegantly bridges spectral graph theory, symbolic dynamics, arithmetic groups, and automata theory, opening new avenues for research at their intersection.