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Ramanujan subshifts

Ievgen Bondarenko, Rostislav Grigorchuk, Alina Vdovina

Abstract

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.

Ramanujan subshifts

Abstract

A finite, connected, -regular graph is called Ramanujan if every its eigenvalue satisfies either or . 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 -regular -subshift of finite type, and we define a Ramanujan subshift as a -regular -subshift with an optimal rate of decay of correlations. We show that for every odd prime power and dimension , there exists a -regular Ramanujan -subshift. The construction is based on the quaternionic lattices over introduced by Rungtanapirom-Stix-Vdovina (2019). Each of our -regular Ramanujan subshifts gives rise to a family of non-bipartite -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 , we get a single lifting rule that can be iterated to produce an infinite family of -regular Ramanujan graphs.
Paper Structure (6 sections, 21 theorems, 62 equations, 11 figures)

This paper contains 6 sections, 21 theorems, 62 equations, 11 figures.

Table of Contents

  1. Introduction
  2. One-dimensional subshifts of finite type
  3. Two-dimensional regular subshifts
  4. VH-datum and regular $\mathbb{Z}^2$-subshifts
  5. Mealy automata and iterated lifts
  6. Quaternionic lattices and Ramanujan subshifts

Key Result

Theorem 1

For every odd prime power $q\geq 3$ and dimension $\delta<q$, there exists a $q$-regular Ramanujan $\mathbb{Z}^\delta$-subshift.

Figures (11)

  • Figure 1: The squares representing relations of the lattice $\Gamma_{1,2}=\langle a,b,x,y\,|\, ax=x^{-1}b^{-1}, ay=xa^{-1}, by=y^{-1}a, bx^{-1}=yb^{-1}\rangle$ in the product $T_4\times T_4$, associated Mealy automaton, deterministic $3$-lifts generating $4$-regular Ramanujan graphs, and Wang tileset. The $3$-regular Ramanujan $\mathbb{Z}^2$-subshift is obtained by forbidding tile color combinations $cc^{-1}$ for $c\in\{a^{\pm 1},b^{\pm 1}, x^{\pm 1}, y^{\pm 1}\}$.
  • Figure 2: Example of a $4$-regular subshift associated to the presentation of the group $F_2\times F_2$
  • Figure 3: Filling two patterns at horizontal distance $n$ and vertical distance $m$
  • Figure 4: Filling two patterns at horizontal distance $n$
  • Figure 5: The tiles corresponding to four tuples in Property (1)
  • ...and 6 more figures

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Corollary 3.1
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Definition 4
  • Definition 5
  • ...and 45 more