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

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 ${\underline{\lim }}_{M\to \infty }{\mu }_2\ge {\overline{\lim }}_{M\to \infty }{\min }_n{\mathcal{H}}_{n,n}$. 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.

5 pages

References in corpus (2)

• Undecidability of the Spectral Gap (short version)
• On the spectra of nonsymmetric Laplacian matrices