拉马努金子移位
Ramanujan subshifts
AI总结：
本文提出了一种高维拉马努金子移位的构造方法，通过四元数格生成非二分的(q+1)-正则拉马努金图，并提供了生成无限家族的提升规则。
摘要：
一个有限的、连通的、(d+1)阶图G被称为拉马努金图，如果其每一个特征值λ满足要么λ=±(d+1)，要么|λ|≤2√d。拉马努金条件对应于与之相关的非回溯边子移位的最优相关衰减率。我们考虑这一观察的高维推广。我们引入了d-正则Z^δ有限型子移位的概念，并定义拉马努金子移位为一种d-正则Z^δ有限型子移位，具有最优的相关衰减率。我们证明对于每一个奇数素数幂q≥3和维度δ<q，存在一个q-正则的拉马努金Z^δ子移位。该构造基于Rungtanapirom-Stix-Vdovina（2019）引入的四元数格 over F_q(t)。每个我们的q-正则拉马努金子移位都会产生一个非二分的(q+1)-正则拉马努金图。这些图非常明确且局部：任何顶点的邻居可以通过与子移位相关的显式梅拉自动机计算。作为附带结果，对于每一个奇数素数幂q，我们得到一个单个提升规则，可以迭代产生无限多个(q+1)-正则拉马努金图。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
