Abstract
We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on \({\mathbb{Z}}^{d}\), which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e\({}^{-{\lambda }_{1}\sqrt{t}}\) for d = 1, as e\({}^{-{\lambda }_{2}t/\log t}\) for d = 2, and as e\({}^{-{\lambda }_{d}t}\) for d ≥ 3, where λ1 and λ2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e\({}^{-\tilde{{\lambda }}_{d}t}\), with \(\tilde{{\lambda }}_{d} > 0\) for all d ≥ 1. A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous time symmetric random walk increases under perturbation by a deterministic path.
AMS 2010 subject classification: 60K37, 60K35, 82C22.
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Notes
- 1.
Note that this is where the proof fails for the γ < 0 case.
References
Antal, P.: Trapping problem for the simple random walk. Dissertation ETH, No 10759 (1994)
Antal, P.: Enlargement of obstacles for the simple random walk. Ann. Probab. 23, 1061–1101 (1995)
Biskup, M., König, W.: Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29, 636–682 (2001)
Bolthausen, E.: Localization of a two-dimensional random walk with an attractive path interaction. Ann. Probab. 22, 875–918 (1994)
Bramson, M., Lebowitz, J.: Asymptotic behavior of densities for two-particle annihilating random walks. J. Statist. Phys. 62, 297–372 (1991)
Cox, T., Griffeath, D.: Large deviations for Poisson systems of independent random walks. Z. Wahrsch. Verw. Gebiete 66, 543–558 (1984)
Donsker, M., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28, 525–565 (1975)
Donsker, M., Varadhan, S.R.S.: On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32, 721–747 (1979)
Evans, L.C.: Partial differential equations, 2nd edn. Graduate Studies in Mathematicsm, vol. 19. American Mathematical Society, Providence, RI (2010)
Feller, W.: An introduction to probability theory and its applications, vol. II. Wiley, New York (1966)
Gärtner, J., den Hollander, F.: Intermittency in a catalytic random medium. Ann. Probab. 34, 2219–2287 (2006)
Gärtner, J., König, W.: The parabolic Anderson model. Interacting Stochastic Systems, pp. 153–179. Springer, Berlin (2005)
Gärtner, J., den Hollander, F., Maillard, G.: Intermittency on catalysts. Trends in Stochastic Analysis, pp. 235–248, London Math. Soc. Lecture Note Ser., vol. 353. Cambridge University Press, Cambridge (2009)
Gärtner, J., den Hollander, F., Maillard, G.: Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment. In: Probability in Complex Physical Systems. Springer, Heidelberg, pp. 159–193 (2012)
Kesten, H., Sidoravicius, V.: Branching random walks with catalysts. Electron. J. Probab. 8, 1–51 (2003)
Kesten, H., Sidoravicius, V.: The spread of a rumor or infection in a moving population. Ann. Probab. 33, 2402–2462 (2005)
Lawler, G.F.: Intersections of Random Walks. Birkhäuser Boston (1996)
Liggett, T.: An improved subadditive ergodic theorem. Ann. Probab. 13, 1279–1285 (1985)
Moreau, M., Oshanin, G., Bénichou, O., Coppey, M.: Pascal principle for diffusion-controlled trapping reactions. Phys. Rev. E 67, 045104(R) (2003)
Moreau, M., Oshanin, G., Bénichou, O., Coppey, M.: Lattice theory of trapping reactions with mobile species. Phys. Rev. E 69, 046101 (2004)
Peres, Y., Sinclair, A., Sousi, P., Stauffer, A.: Mobile geometric graphs: Detection, coverage and percolation. Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (50DA), 412–428 (2011)
Redig, F.: An exponential upper bound for the survival probability in a dynamic random trap model. J. Stat. Phys. 74, 815–827 (1994)
Spitzer, F.: Principles of Random Walk, 2nd edn. Springer, Berlin (1976)
Sznitman, A.S.: Brownian Motion, Obstacles and Random Media. Springer, Berlin (1998)
Varadhan, S.R.S.: Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56, 1222–1245 (2003)
Acknowledgements
We thank Frank den Hollander for bringing [22] to our attention, Alain-Sol Sznitman for suggesting that we prove a shape theorem for the quenched survival probability, and Vladas Sidoravicius for explaining to us [16, Prop. 8], which we use to prove the positivity of the quenched Lyapunov exponent. A.F. Ramírez was partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico grant 1100298. J. Gärtner, R. Sun, and partially A.F. Ramírez were supported by the DFG Forschergruppe 718 Analysis and Stochastics in Complex Physical Systems.
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Drewitz, A., Gärtner, J., Ramírez, A.F., Sun, R. (2012). Survival Probability of a Random Walk Among a Poisson System of Moving Traps. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_6
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