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

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 654-697.

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.

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.

[1] V. Ambegoakar, B. Halperin and J. S. Langer. Hopping conductivity in disordered systems. Phys. Rev. B 4 (1971) 2612-2620.

[2] A. Miller and E. Abrahams. Impurity conduction at low concentrations. Phys. Rev. 120 (1960) 745-755. | Zbl

[3] P. Antal and A. Pisztora. On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996) 1036-1048. | MR | Zbl

[4] N. W. Ashcroft and N. D. Mermin. Solid State Phyisics. Saunders College, Philadelphia, 1976.

[5] M. T. Barlow. Random walks on supercritical percolation clusters. Ann. Probab. 32 (4) (2004) 3024-3084. | MR | Zbl

[6] M. T. Barlow and H.-D. Deuschel. Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38 (2010) 234-276. | MR | Zbl

[7] N. Berger and M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (1-2) (2007) 83-120. | MR | Zbl

[8] N. Berger, M. Biskup, C. E. Hoffman and G. Kozma. Anomalous heat-kernel decay for random walk among bounded random conductances. Ann. Inst. Henri Poincaré Probab. Stat. 274 (2008) 374-392. | Numdam | MR | Zbl

[9] M. Biskup and T. Prescott. Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12 (2007) 1323-1348. | MR | Zbl

[10] D. Boivin. Weak convergence for reversible random walks in a random environment. Ann. Probab. 21 (1993) 1427-1440. | MR | Zbl

[11] D. Boivin and J. Depauw. Spectral homogenization of reversible random walks on ${\mathbb{Z}}^d$ in a random environment. Stochastic Process. Appl. 104 (2003) 29-56. | MR | Zbl

[12] P. Caputo and A. Faggionato. Isoperimetric inequalities and mixing time for a random walk on a random point process. Ann. Appl. Probab. 17 (2007) 1707-1744. | MR | Zbl

[13] P. Caputo and A. Faggionato. Diffusivity in one-dimensional generalized Mott variable-range hopping models. Ann. Appl. Probab. 19 (4) (2009) 1459-1494. | MR | Zbl

[14] P. Caputo, A. Faggionato and A. Gaudilliere. Recurrence and transience for long-range reversible random walks on a random point process. Electron. J. Probab. 14 (2009) 2580-2616. | MR | Zbl

[15] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Springer, New York, 1988. | MR | Zbl

[16] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. Invariance principle for reversible Markov processes with applications to random motions in random environments. J. Stat. Phys. 55 (1989) 787-855. | MR | Zbl

[17] R. Durrett. Probability: Theory and Examples, 3rd edition. Thomson, Cambridge, 2005. | MR | Zbl

[18] A. Faggionato and P. Mathieu. Mott law for Mott variable-range random walk. Comm. Math. Phys. 281 (1) (2008) 263-286. | MR | Zbl

[19] A. Faggionato, H. Schulz-Baldes and D. Spehner. Mott law as lower bound for a random walk in a random environment. Comm. Math. Phys. 263 (2006) 21-64. | MR | Zbl

[20] L. R. G. Fontes and P. Mathieu. On symmetric random walks with random conductances on $Z^d$. Probab. Theory Related Fields 134 (2006) 565-602. | MR | Zbl

[21] G. Grimmett. Percolation, 2nd edition. Grundlehren 321. Springer, Berlin, 1999. | MR

[22] C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov process and applications to simple exclusion. Comm. Math. Phys. 104 (1986) 1-19. | MR | Zbl

[23] S. M. Kozlov. The method of averaging and walks in inhomogeneous environments. Russian Math. Surveys 40 (2) (1985) 73-145. | MR | Zbl

[24] T. M. Liggett, R. H. Schonmann and A. M. Stacey. Domination by product measures. Ann. Probab. 25 (1997) 71-95. | MR | Zbl

[25] P. Mathieu. Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130 (2008) 1025-1046. | MR | Zbl

[26] P. Mathieu and A. Piatniski. Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007) 2287-2307. | MR | Zbl

[27] B. Morris and Y. Peres. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2005) 245-266. | MR | Zbl

[28] N. F. Mott. Conduction in glasses containing transition metal ions. J. Non-Crystal. Solids 1 (1968) 1.

[29] N. F. Mott. Conduction in non-crystalline materials. Phil. Mag. 19 (1969) 835.

[30] N. F. Mott. Electrons in glass. In Nobel Lectures, Physics 1971-1980. World Scientific, Singapore, 1992.

[31] N. F. Mott and E. A. Davis. Electronic Processes in Non-Crystaline Materials. Oxford Univ. Press, Oxford, 1979. | Zbl

[32] V. Sidoravicius and A. S. Sznitman. Quenched invariance principles for walks on clusters of percolations or among random conductances. Probab. Theory Related Fields 129 (2) (2004) 219-244. | MR | Zbl

[33] B. Shklovskii and A. L. Efros. Electronic Properties of Doped Semiconductors. Springer, Berlin, 1984.

[34] H. Spohn. Large Scale Dynamics of Interacting Particles. Springer, Berlin, 1991. | Zbl