Spectral Analysis of 1d Nearest–neighbor Random Walks with Applications to Subdiffusive Random Trap and Barrier Models

Given a family X(t) of continuous–time nearest–neighbor random walks on the one dimensional lattice Z, parameterized by n ∈ N+, we show that the spectral analysis of the Markov generator of X with Dirichlet conditions outside (0, n) reduces to the analysis of the eigenvalues and eigenfunctions of a suitable generalized second order differential operator −DmnDx with Dirichlet conditions outside (0, 1). If in addition the measures dmn weakly converge to some measure dm, similarly to Krein’s correspondence we prove a limit theorem of the eigenvalues and eigenfunctions of −DmnDx to the corresponding spectral quantities of −DmDx. Applying the above result together with the Dirichlet–Neumann bracketing, we investigate the limiting behavior of the small eigenvalues of subdiffusive random trap and barrier models and establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions.