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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.

MSC:
 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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