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

Given a Hermitian matrix with possibly degenerate eigenvalues , we provide, in the limit M → ∞, a lower bound for the gap assuming that i) the eigenvector (eigenvectors) associated to is ergodic (are all ergodic) and ii) the off-diagonal terms of vanish for M → ∞. Under these hypotheses, we find . 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 . 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.