Upper tails of self-intersection local times of random walks: survey of proof techniques

Wolfgang König

doi:10.5802/acirm.18

Wolfgang König ¹

¹ Technical University Berlin, Str. des 17. Juni 136, 10623 Berlin, and Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Actes des rencontres du CIRM, Volume 2 (2010) no. 1, pp. 15-24.

Abstract

The asymptotics of the probability that the self-intersection local time of a random walk on $ℤ^{d}$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated techniques and results. This is an extended summary of a talk held on the CIRM-conference on Excess self-intersection local times, and related topics in Luminy, 6-10 Dec., 2010.

Published online: 2011-01-02

Zbl: 06938566

DOI: 10.5802/acirm.18

Classification: 60K37, 60F10, 60J55
Keywords: Self-intersection local time, upper tail, Donsker-Varadhan large deviations, variational formula

Author's affiliations:

Wolfgang König ¹

¹ Technical University Berlin, Str. des 17. Juni 136, 10623 Berlin, and Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

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Wolfgang König. Upper tails of self-intersection local times of random walks: survey of proof techniques. Actes des rencontres du CIRM, Volume 2 (2010) no. 1, pp. 15-24. doi : 10.5802/acirm.18. https://pmb.cedram.org/articles/10.5802/acirm.18/

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