Electronic Communications in Probability

From the Lifshitz tail to the quenched survival asymptotics in the trapping problem

Ryoki Fukushima

Full-text: Open access


The survival problem for a diffusing particle moving among random traps is considered. We introduce a simple argument to derive the quenched asymptotics of the survival probability from the Lifshitz tail effect for the associated operator. In particular, the upper bound is proved in fairly general settings and is shown to be sharp in the case of the Brownian motion among Poissonian obstacles. As an application, we derive the quenched asymptotics for the Brownian motion among traps distributed according to a random perturbation of the lattice.

Article information

Electron. Commun. Probab., Volume 14 (2009), paper no. 42, 435-446.

Accepted: 6 October 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Trapping problem random media survival probability Lifshitz tail

This work is licensed under aCreative Commons Attribution 3.0 License.


Fukushima, Ryoki. From the Lifshitz tail to the quenched survival asymptotics in the trapping problem. Electron. Commun. Probab. 14 (2009), paper no. 42, 435--446. doi:10.1214/ECP.v14-1497. https://projecteuclid.org/euclid.ecp/1465234751

Export citation


  • Antal, Peter. Enlargement of obstacles for the simple random walk. Ann. Probab. 23 (1995), no. 3, 1061–1101.
  • Biskup, Marek; König, Wolfgang. Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29 (2001), no. 2, 636–682.
  • Carmona, René; Lacroix, Jean. Spectral theory of random Schrödinger operators. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1990. xxvi+587 pp. ISBN: 0-8176-3486-X
  • den Hollander, Frank; Weiss, George H. Aspects of Trapping in Transport Processes. In Contemporary Problems in Statistical Physics, 147–203. Society for Industrial and Applied Mathematics, 1994.
  • Donsker, M. D.; Varadhan, S. R. S. Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28 (1975), no. 4, 525–565. (53 #1757a)
  • Fukushima, Masatoshi. On the spectral distribution of a disordered system and the range of a random walk. Osaka J. Math. 11 (1974), 73–85.
  • Fukushima, Ryoki. Brownian survival and Lifshitz tail in perturbed lattice disorder. J. Funct. Anal. 256 (2009), no. 9, 2867–2893.
  • Fukushima, Ryoki; Ueki, Naomasa. Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice. Kyoto University Preprint Series, Kyoto-Math 2009-10, submitted, 2009.
  • Havlin, Shlomo; Ben-Avraham, Daniel. Diffusion in disordered media. Adv. Phys., 36(6):695–798, 1987.
  • Kasahara, Yuji. Tauberian theorems of exponential type. J. Math. Kyoto Univ. 18 (1978), no. 2, 209–219.
  • Lifshitz, I. M. Energy spectrum structure and quantum states of disordered condensed systems. Uspehi Fiz. Nauk 83 617–663 (Russian); translated as Soviet Physics Uspekhi 7 1965 549–573.
  • Nakao, Shintaro. On the spectral distribution of the Schrödinger operator with random potential. Japan. J. Math. (N.S.) 3 (1977), no. 1, 111–139.
  • Pastur, L. A. The behavior of certain Wiener integrals as $t \rightarrow \infty$ and the density of states of Schrödinger equations with random potential. (Russian) Teoret. Mat. Fiz. 32 (1977), no. 1, 88–95.
  • Reed, Michael; Simon, Barry. Methods of modern mathematical physics. I. Functional analysis. Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. xv+400 pp. ISBN: 0-12-585050-6
  • Romerio, M.; Wreszinski, W. On the Lifschitz singularity and the tailing in the density of states for random lattice systems. J. Statist. Phys. 21 (1979), no. 2, 169–179.
  • Seneta, Eugene. Regularly varying functions. Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin-New York, 1976. v+112 pp.
  • Simon, Barry. Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526.
  • Sznitman, Alain-Sol. Brownian asymptotics in a Poissonian environment. Probab. Theory Related Fields 95 (1993), no. 2, 155–174.
  • 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
  • van den Berg, M. A Gaussian lower bound for the Dirichlet heat kernel. Bull. London Math. Soc. 24 (1992), no. 5, 475–477.