Long-range expanders: construction and cutoff

Abstract

Long-range expansion is a tree-like volume growth condition on bounded-degree regular graphs, introduced by Dodos, Tikhomirov, Tyros, and the author to prove quantitative nonlinear Poincaré inequalities and non-embedding theorems. Motivated by fundamental questions in geometric group theory and nonlinear functional analysis about superexpanders of logarithmic girth, our first main result is that every Ramanujan graph has long-range expansion. This gives explicit long-range expanders with logarithmic girth (such as Lubotzky--Phillips--Sarnak graphs) and establishes a strict hierarchy: Ramanujan expansion implies long-range expansion, which implies spectral expansion, with neither converse holding. We subsequently compare the different notions of expansion from the perspective of dynamics. Our second main result establishes cutoff with an explicit Gaussian limit profile for the random walk on every fixed-degree long-range expander sequence. In particular, the cutoff location and profile coincide with those for Ramanujan graphs. This offers intermediate progress between cutoff for Ramanujan graphs---proven by Lubetzky and Peres---and the long-standing conjecture of cutoff for transitive spectral expanders.