Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues

Abstract

It was recently shown in [23] and [27] that Ramanujan graphs, i.e., graphs with the optimal spectrum, exhibit cutoff of the simple random walk in optimal time and have optimal almost-diameter. We prove that this spectral condition can be replaced by a weaker condition, the Sarnak-Xue density of eigenvalues property, to deduce similar results.We show that a family of Schreier graphs of the SL2 (Ft)-action on the projective line satisfies the Sarnak-Xue density condition, and hence exhibit the desired properties. To the best of our knowledge, this is the first known example of optimal cutoff and almost-diameter on an explicit family of graphs that are neither random nor Ramanujan.

Bounded cutoff window for the non-backtracking random walk on Ramanujan Graphs

“…Golubev and Kamber [16] show that the density together with the assumption that X is an expander suffice to show that the SRW on X obeys cutoff at rate p+1 p−1 log p n. We show that the shortest possible cutoff applies to the NBRW. Theorem 1.4.…”

“…For a discussion of this density hypothesis see [16]. The point is that this density can often be established in cases where the Ramanujan is not known or even fails.…”

On Sarnak's Density Conjecture and its Applications

“…The details are given in Subsection 4.4. We remark that when the graphs are regular the proofs are also given in [26].…”

“…We remark that it seems that the problem is harder for principal congruence subgroups, and easier for group that are far from being normal, such as Γ 0 (N ) of SL m (Z) (see Subsection 2.5). See, for example, the Density Amplification phenomena for graphs in [26].…”

Small diameters and generators for arithmetic lattices in $\mathrm{SL}_2(\mathbb{R})$ and certain Ramanujan graphs

“…These Ramanujan graphs (by definition) enjoy a large spectral gap, which in turn yields a small upper bound on the diameter. The incorporation of a density estimate for large eigenvalues for (homogeneous) expander graphs has proved valuable in showing that they admit a smaller diameter than what could be directly inferred from their spectral gap [9]. In Section §4, we demonstrate the usefulness of the fourth moment bound of Khayutin-Nelson-Steiner [15] also in the context of expander graphs, by proving upper bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound which are of equal strength.…”