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Diffusivity in onedimensional generalized Mott variablerange hopping models. Available at arXiv:math.PR/0701253
, 2007
"... We consider random walks in a random environment which are generalized versions of wellknown effective models for Mott variablerange hopping. We study the homogenized diffusion constant of the random walk in the onedimensional case. We prove various estimates on the lowtemperature behavior which ..."
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Cited by 4 (2 self)
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We consider random walks in a random environment which are generalized versions of wellknown effective models for Mott variablerange hopping. We study the homogenized diffusion constant of the random walk in the onedimensional case. We prove various estimates on the lowtemperature behavior which confirm and extend previous work by physicists. 1. Introduction. Random
Mott law as upper bound for a random walk in a random environment
"... Abstract. We consider a random walk on the support of an ergodic simple point process on R d, d≥2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model ..."
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Cited by 3 (1 self)
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Abstract. We consider a random walk on the support of an ergodic simple point process on R d, d≥2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8]. Key words: disordered system, Mott law, random walk in random environment, marked point process, stochastic domination, continuum percolation. 1.
Mott law for Mott variablerange random walk
 Comm. Math. Phys
, 2008
"... Abstract. We consider a random walk on the support of an ergodic simple point process on R d, d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective mode ..."
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Cited by 2 (2 self)
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Abstract. We consider a random walk on the support of an ergodic simple point process on R d, d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process we prove an upper bound of the asymptotic diffusion matrix of the random walk in agreement with Mott law. A lower bound in agreement with Mott law was proved in [6]. Key words: disordered systems, random walk in random environment, marked point process, stochastic domination, continuum percolation.
RECURRENCE AND TRANSIENCE FOR LONG–RANGE REVERSIBLE RANDOM WALKS ON A RANDOM POINT PROCESS
, 811
"... Abstract. We consider reversible random walks in random environment obtained from symmetric long–range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obta ..."
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Abstract. We consider reversible random walks in random environment obtained from symmetric long–range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network. Key words: random walk in random environment, recurrence, transience, point process,
INVARIANCE PRINCIPLE FOR MOTT VARIABLE RANGE HOPPING AND OTHER WALKS ON POINT PROCESSES
"... Abstract. We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the αpower of the jump length and depend on the energy marks via a Boltzmann–like factor. The case α = 1 corresponds to the phononinduced Mott variable range hopping ..."
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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 αpower of the jump length and depend on the energy marks via a Boltzmann–like factor. The case α = 1 corresponds to the phononinduced 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. Key words: random walk in random environment, Poisson point process, percolation, stochastic domination, invariance principle, corrector.