Nonlinear spectral calculus and super-expanders

TL;DR

This paper develops a nonlinear spectral calculus framework demonstrating how spectral gaps decay and behave under graph operations, leading to a new combinatorial construction of super-expanders that resist embedding into uniformly convex spaces.

Contribution

It introduces a spectral calculus inequality for nonlinear spectral gaps and shows their sub-multiplicative behavior under zigzag products, enabling super-expander construction.

Findings

Spectral gaps decay along Cesaro averages.

Nonlinear spectral gaps are sub-multiplicative under zigzag products.

Constructs super-expanders that cannot embed into uniformly convex spaces.

Abstract

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.