Uniqueness of the mixing measure for a random walk in a random environment on the positive integers.
Upper bound of a volume exponent for directed polymers in a random environment
Upper tails of self-intersection local times of random walks: survey of proof techniques
The asymptotics of the probability that the self-intersection local time of a random walk on 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...