Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models. (English) Zbl 1248.60105

Summary: We consider a sequence \(X^{(n)}, n \geq 1 \), of continuous-time nearest-neighbor random walks on the one dimensional lattice \(\mathbb{Z}\). 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 giveninterval. If the measures \(dm_n\) weakly converge to some measure \(dm_\infty\), we prove a limit theorem for the eigenvalues and eigenfunctions of \(-D_{m_n}D_x\) to the corresponding spectral quantities of \(-D_{m_\infty} D_x\). 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.


60K37 Processes in random environments
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
34B24 Sturm-Liouville theory
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