Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Invariance principle for Mott variable range hopping and other walks on point processes

P. Caputo, A. Faggionato, and T. Prescott

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Abstract

We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the $\alpha$-power of the jump length and depend on the energy marks via a Boltzmann-like factor. The case $\alpha=1$ corresponds to the phonon-induced Mott variable range hopping in disordered solids in the regime of strong Anderson localization. We prove that for almost every realization of the marked process, the diffusively rescaled random walk, with an arbitrary start point, converges to a Brownian motion whose diffusion matrix is positive definite and independent of the environment. Finally, we extend the above result to other point processes including diluted lattices.

Résumé

On considère une marche aléatoire sur les points d’un processus de Poisson marqué. Les taux de saut ont une décroissance exponentielle en fonction de la longueur du saut, généralisant le modèle de sauts à portée variable de Mott pour les systèmes désordonnés en regime de localisation forte d’Anderson. On montre que pour presque toute réalisation du processus ponctuel marqué, la marche aléatoire de point de départ arbitraire satisfait un principe d’invariance avec matrice de diffusion limite déterministe définie positive. On montre que ce resultat s’étend à d’autres processus ponctuels incluant les réseaux dilués.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 3 (2013), 654-697.

Dates
First available in Project Euclid: 2 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1372772640

Digital Object Identifier
doi:10.1214/12-AIHP490

Mathematical Reviews number (MathSciNet)
MR3112430

Zentralblatt MATH identifier
1296.60268

Subjects
Primary: 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles 60G55: Point processes

Keywords
Random walk in random environment Poisson point process Percolation Stochastic domination Invariance principle Corrector

Citation

Caputo, P.; Faggionato, A.; Prescott, T. Invariance principle for Mott variable range hopping and other walks on point processes. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 3, 654--697. doi:10.1214/12-AIHP490. https://projecteuclid.org/euclid.aihp/1372772640


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