Uniform estimates of nonlinear spectral gaps

TL;DR

This paper establishes uniform lower bounds for nonlinear spectral gaps of finite graphs, independent of the target metric space, and applies these results to trees and Hamming cubes to analyze their asymptotic behavior.

Contribution

It generalizes the path method to prove uniform bounds on nonlinear spectral gaps, revealing metric space independence and asymptotic sharpness for specific graph families.

Findings

Nonlinear spectral gaps are uniformly bounded below regardless of the target metric space.

Asymptotic behavior of gaps in regular trees is metric space independent.

Estimate of the nonlinear spectral gap for Hamming cubes is asymptotically sharp.

Abstract

By generalizing the path method, we show that nonlinear spectral gaps of a finite connected graph are uniformly bounded from below by a positive constant which is independent of the target metric space. We apply our result to an $r$-ball $T_{d,r}$ in the $d$-regular tree, and observe that the asymptotic behavior of nonlinear spectral gaps of $T_{d,r}$ as $r\to\infty$ does not depend on the target metric space, which is in contrast to the case of a sequence of expanders. We also apply our result to the $n$-dimensional Hamming cube $H_n$ and obtain an estimate of its nonlinear spectral gap with respect to an arbitrary metric space, which is asymptotically sharp as $n\to\infty$.