Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

TL;DR

This paper analyzes the spectral properties of 1D nearest-neighbor random walks and their limits, providing new insights into eigenvalues, eigenfunctions, and applications to subdiffusive trap and barrier models.

Contribution

It introduces a spectral analysis framework linking random walk generators to differential operators and establishes convergence and bounds for eigenvalues in these models.

Findings

Eigenvalues and eigenfunctions converge under measure limits.

Dirichlet-Neumann bracketing provides bounds for eigenvalue counting.

Applications to subdiffusive trap and barrier models in physics.

Abstract

We consider a family X^{(n)}, n \in \bbN_+, of continuous-time nearest-neighbor random walks on the one dimensional lattice Z. We reduce the spectral analysis of the Markov generator of X^{(n)} with Dirichlet conditions outside (0,n) to the analogous problem for a suitable generalized second order differential operator -D_{m_n} D_x, with Dirichlet conditions outside a given interval. If the measures dm_n weakly converge to some measure dm_*, we prove a limit theorem for the eigenvalues and eigenfunctions of -D_{m_n}D_x to the corresponding spectral quantities of -D_{m_*} D_x. As second result, we prove the Dirichlet-Neumann bracketing for the operators -D_m D_x and, as a consequence, we establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions in the case that m is a self--similar stochastic process. Finally, we apply the above results to investigate the spectral structure of some classes of subdiffusive random trap and barrier models coming from one-dimensional physics.