Uniqueness of the second eigenspace of the interchange process

TL;DR

This paper characterizes the second eigenspace of the interchange process, showing it is uniquely determined by the second eigenvectors of the random walk on most graphs, with a notable exception.

Contribution

It provides a detailed characterization of the second eigenspace of the interchange process, extending spectral gap results to a broader understanding of eigenvector structure.

Findings

The second eigenspace is uniquely determined by the random walk eigenvectors on all connected graphs except the 4-cycle.

The proof uses an induction scheme, the octopus inequality, and graph Laplacian computations.

The spectral gap theorem applies broadly with a specific exception for the 4-cycle.

Abstract

The spectral gap theorem of Caputo, Liggett, and Richthammer states that on any connected graph equipped with edge weights, the 2nd eigenvalue of the interchange process equals the 2nd eigenvalue of the random walk process. In this work we characterize the 2nd eigenspace of the interchange process. We prove that this eigenspace is uniquely determined by the 2nd eigenvectors of the random walk process on every connected weighted graph except the $4$-cycle with uniform edge weights. The key to our proof is an induction scheme on the number of vertices, and involves the octopus (in)equality, representation theoretic computations, and graph Laplacian computations.