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

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.