A combinatorial approach to nonlinear spectral gaps
Dylan J. Altschuler, Pandelis Dodos, Konstantin Tikhomirov, Konstantinos Tyros
TL;DR
The paper addresses the challenge of extending discrete Poincaré inequalities from scalar functions to vector-valued functions in normed spaces with nontrivial cotype, focusing on regular graphs. It introduces long-range expansion as a key combinatorial property and proves that any d-regular graph with this property satisfies a nonlinear Poincaré inequality into spaces with an unconditional basis, with a Poincaré constant that grows only polynomially in the space's cotype q (and linearly in the exponent p for order-$p$ norms). The results yield a near-optimal dependence on q and imply that random regular graphs require cotype at least polylogarithmic in the graph size for low-distortion embeddings into such normed spaces. The work also outlines a program for extending these bounds to Banach lattices, refining the exponent on q, and constructing explicit graphs with long-range expansion, linking graph geometry with nonlinear functional-analytic properties and embedding theory.
Abstract
A seminal open question of Pisier and Mendel--Naor asks whether every degree-regular graph which satisfies the classical discrete Poincaré inequality for scalar functions, also satisfies an analogous inequality for functions taking values in \textit{any} normed space with non-trivial cotype. Motivated by applications, it is also greatly important to quantify the dependence of the corresponding optimal Poincaré constant on the cotype $q$. Works of Odell--Schlumprecht (1994), Ozawa (2004), and Naor (2014) make substantial progress on the former question by providing a positive answer for normed spaces which also have an unconditional basis, in addition to finite cotype. However, little is known in the way of quantitative estimates: the mentioned results imply a bound on the Poincaré constant depending super-exponentially on $q$. We introduce a novel combinatorial framework for proving quantitative nonlinear spectral gap estimates. The centerpiece is a property of regular graphs that we call \emph{long range expansion}, which holds with high probability for random regular graphs. Our main result is that any regular graph with the long-range expansion property satisfies a discrete Poincaré inequality for any normed space with an unconditional basis and cotype $q$, with a Poincaré constant that depends \emph{polynomially} on $q$, which is optimal. As an application, any normed space with an unconditional basis which admits a low distortion embedding of an $n$-vertex random regular graph, must have cotype at least polylogarithmic in $n$. This extends a celebrated lower-bound of Matoušek for low distortion embeddings of random graphs into $\ell_q$ spaces.