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

¹ Departamento de Fisica, Universidade Federal de Santa Catarina - Florianopolis, 88040-900, SC, Brazil
² Dipartimento di Fisica, Università di Roma La Sapienza - Piazzale Aldo Moro 2, Roma I-00185, Italy
³ Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1 - Roma 00185, Italy

Received: 20 December 2015
Accepted: 28 January 2016

Abstract

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.