Asymptotic lower bound for the gap of Hermitian matrices having ergodic ground states and infinitesimal off-diagonal elements

TL;DR

This paper establishes a lower bound on the spectral gap of large Hermitian matrices with ergodic ground states and infinitesimal off-diagonal elements, with implications for relaxation times in stochastic processes.

Contribution

It provides a novel asymptotic lower bound for the spectral gap of large Hermitian matrices under specific ergodicity and decay conditions, linking spectral properties to graph degrees.

Findings

Lower bound for the spectral gap in the limit of large matrix size.

The gap is bounded below by the minimal diagonal element in the limit.

Application to symmetric random walks shows relaxation time is bounded by minimal degree nodes.

Abstract

Given a $M\times M$ Hermitian matrix $\mathcal{H}$ with possibly degenerate eigenvalues $\mathcal{E}_1 < \mathcal{E}_2 < \mathcal{E}_3< \dots$, we provide, in the limit $M\to\infty$, a lower bound for the gap $\mu_2 = \mathcal{E}_2 - \mathcal{E}_1$ assuming that (i) the eigenvector (eigenvectors) associated to $\mathcal{E}_1$ is ergodic (are all ergodic) and (ii) the off-diagonal terms of $\mathcal{H}$ vanish for $M\to\infty$ more slowly than $M^{-2}$. Under these hypotheses, we find $\varliminf_{M\to\infty} \mu_2 \geq \varlimsup_{M\to\infty} \min_{n} \mathcal{H}_{n,n}$. This general result turns out to be important for upper bounding the relaxation time of linear master equations characterized by a matrix equal, or isospectral, to $\mathcal{H}$. As an application, we consider symmetric random walks with infinitesimal jump rates and show that the relaxation time is upper bounded by the configurations (or nodes) with minimal degree.