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

Givena M M Hermitian matrix H with possibly degenerate eigenvalues E11<E2<E3<... , we provide, in the limit M →∞, a lower bound for the gap μ 2=E2−E1assum-ing that i) the eigenvector (eigenvectors) associated to E1is ergodic (are all ergodic) and ii) theoff-diagonal terms of H vanish for M→∞. Under these hypotheses, we find lim M→∞2 ≥ limM→∞minnHn,n. This general result turns out to be important for upper bounding the relax- ation time of linear master equations characterized by a matrix equal, or isospectral, to 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 degre