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Register a new userMixing properties and the chromatic number of Ramanujan complexes
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Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application we prove that non-partite Ramanujan complexes have high girth and high chromatic number, generalizing a well known result about Ramanujan graphs.
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The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $G$ over a local field $F$. We show that if $T$ is any $k$-regular $G$-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the $n$-vertex Ramanujan complex has cutoff at time $log_k n$. The high dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of $G$. Via these, we show that operators $T$ as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property ($r$-normal), implying cutoff. Applications include geodesic flow operators, geometric implications, and a confirmation of the Riemann Hypothesis for the associated zeta functions over every group $G$, previously known for groups of type $widetilde A_n$ and $widetilde C_2$.
Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments are surveyed. After explaining their connection to the Ramanujan conjecture we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to golden gates which are of importance in quantum computation.
We introduce new methods for understanding the topology of $Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $Hom(T,-)$ and $Hom(-,G)$ as functors from graphs to posets, and introduce a functor $(-)^1$ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of $Hom$ complexes in terms of spaces of equivariant poset maps and $Gamma$-twisted products of spaces. When $P = F(X)$ is the face poset of a simplicial complex $X$, this provides a useful way to control the topology of $Hom$ complexes. Our foremost application of these results is the construction of new families of `test graphs with arbitrarily large chromatic number - graphs $T$ with the property that the connectivity of $Hom(T,G)$ provides the best possible lower bound on the chromatic number of $G$. In particular we focus on two infinite families, which we view as higher dimensional analogues of odd cycles. The family of `spherical graphs have connections to the notion of homomorphism duality, whereas the family of `twisted toroidal graphs lead us to establish a weakened version of a conjecture (due to Lov{a}sz) relating topological lower bounds on chromatic number to maximum degree. Other structural results allow us to show that any finite simplicial complex $X$ with a free action by the symmetric group $S_n$ can be approximated up to $S_n$-homotopy equivalence as $Hom(K_n,G)$ for some graph $G$; this is a generalization of a result of Csorba. We conclude the paper with some discussion regarding the underlying categorical notions involved in our study.
Ramanujan graphs have extremal spectral properties, which imply a remarkable combinatorial behavior. In this paper we compute the high dimensional Hodge-Laplace spectrum of Ramanujan triangle complexes, and show that it implies a combinatorial expansion property, and a pseudo-randomness result. For this purpose we prove a Cheeger-type inequality and a mixing lemma of independent interest.
We determine the asymptotic behaviour of the chromatic number of exchangeable random graphs defined by step-regulated graphons. Furthermore, we show that the upper bound holds for a general graphon. We also extend these results to sparse random graphs obtained by percolations on graphons.
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