## Electronic Journal of Probability

### Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances Model

#### Abstract

We study models of continuous-time, symmetric random walks in random environment on the d-dimensional integer lattice, driven by a field of i.i.d random nearest-neighbor conductances bounded only from above with a power law tail near 0. We are interested in estimating the quenched asymptotic behavior of the on-diagonal heat-kernel. We show that the spectral dimension is standard when we lighten sufficiently the tails of the conductances. As an expected consequence, the same result holds for the discrete-time case.

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 68, 2069-2086.

Dates
Accepted: 8 December 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819853

Digital Object Identifier
doi:10.1214/EJP.v15-839

Mathematical Reviews number (MathSciNet)
MR2745726

Zentralblatt MATH identifier
1231.60037

Rights

#### Citation

Boukhadra, Omar. Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances Model. Electron. J. Probab. 15 (2010), paper no. 68, 2069--2086. doi:10.1214/EJP.v15-839. https://projecteuclid.org/euclid.ejp/1464819853

#### References

• Barlow, Martin T. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004), no. 4, 3024–3084.
• Berger, Noam; Biskup, Marek. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007), no. 1-2, 83–120.
• Berger, N.; Biskup, M.; Hoffman, C. E.; Kozma, G. Anomalous heat-kernel decay for random walk among bounded random conductances. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 2, 374–392.
• Biskup, Marek; Prescott, Timothy M. Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12 (2007), no. 49, 1323–1348.
• Boukhadra, Omar. Heat-kernel estimates for random walk among random conductances with heavy tail. Stochastic Process. Appl. 120 (2010), no. 2, 182–194.
• Delmotte, Thierry. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 (1999), no. 1, 181–232.
• De Masi, A.; Ferrari, P. A.; Goldstein, S.; Wick, W. D. Invariance principle for reversible Markov processes with application to diffusion in the percolation regime. Particle systems, random media and large deviations (Brunswick, Maine, 1984), 71–85, Contemp. Math., 41, Amer. Math. Soc., Providence, RI, 1985.
• De Masi, A.; Ferrari, P. A.; Goldstein, S.; Wick, W. D. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 (1989), no. 3-4, 787–855.
• Fontes, L. R. G.; Mathieu, P. On symmetric random walks with random conductances on ${Bbb Z}sp d$. Probab. Theory Related Fields 134 (2006), no. 4, 565–602.
• Grimmett, Geoffrey. Percolation.Second edition.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6.
• Heicklen, Deborah; Hoffman, Christopher. Return probabilities of a simple random walk on percolation clusters. Electron. J. Probab. 10 (2005), no. 8, 250–302 (electronic).
• Horn, Roger A.; Johnson, Charles R. Matrix analysis.Cambridge University Press, Cambridge, 1985. xiii+561 pp. ISBN: 0-521-30586-1.
• P. Mathieu, Quenched invariance principles for random walks with random conductances, J. Stat. Phys. 130 (2008), no. 5, 1025–1046.
• Mathieu, P.; Piatnitski, A. Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2085, 2287–2307.
• Mathieu, Pierre; Remy, Elisabeth. Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004), no. 1A, 100–128.
• Sidoravicius, Vladas; Sznitman, Alain-Sol. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2004), no. 2, 219–244.
• Sznitman, Alain-Sol. Brownian motion, obstacles and random media.Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xvi+353 pp. ISBN: 3-540-64554-3.