Upper tails of self-intersection local times of random walks: survey of proof techniques. (English) Zbl 1474.60122
Summary: The asymptotics of the probability that the self-intersection local time of a random walk on \(\mathbb Z^d\) 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.
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60J55 | Local time and additive functionals |
| 60K37 | Processes in random environments |
| 60F10 | Large deviations |
Keywords:
self-intersection local time; upper tail; Donsker-Varadhan large deviations; variational formulaReferences:
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