WIAS Preprint No. 1568, (2010)

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


  • König, Wolfgang
    ORCID: 0000-0002-7673-4364

2010 Mathematics Subject Classification

  • 60K37 60F10 60J55


  • Self-intersection local time, upper tail, Donsker-Varadhan large deviations, variational formula




We discuss the logarithmic asymptotics for the upper tails of self-intersection local times of random walks on Zd. This topic has been studied a lot in the last decade, since it is a natural question, and a rich phenemonology of critical behaviours of the random walk arises, depending on the dimension, the intersection parameter, the scale, and the type of the random process. Furthermore, the question is technically difficult to handle, due to bad continuity and boundedness properties of the self-intersection local time. A couple of different techniques for studying self-intersections have been introduced yet, wich turned out to be more or less fruitful in various situations. It is the goal of this note to survey and compare some of the most fruitful techniques used in recent years.

Appeared in

  • Excess Self-Intersections & Related Topics, vol. 2 of Actes des Rencontres du CIRM, Centre International de Rencontres Mathématiques, Marseille, 2010, pp. 15--24

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