Ramanujan subshifts
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912927554994176 |
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| author | Bondarenko, Ievgen Grigorchuk, Rostislav Vdovina, Alina |
| author_facet | Bondarenko, Ievgen Grigorchuk, Rostislav Vdovina, Alina |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22356 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ramanujan subshifts Bondarenko, Ievgen Grigorchuk, Rostislav Vdovina, Alina Dynamical Systems Combinatorics Group Theory 37B51, 37A25, 05C48 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. |
| title | Ramanujan subshifts |
| topic | Dynamical Systems Combinatorics Group Theory 37B51, 37A25, 05C48 |
| url | https://arxiv.org/abs/2602.22356 |