We study paths of time-length @math of a continuous-time random walk on @math subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The interaction enters through a Gibbs weight at inverse temperature @math ; the "energy" is the total sum of the edge weights for edges on the outer boundary of the range. For edge weights sampled from a translation-invariant, ergodic law, we prove that the range boundary condensates around an asymptotic shape in the limit @math followed by @math . The limit shape is a minimizer (unique, modulo translates) of the sum of the principal harmonic frequency of the domain and the perimeter with respect to the first-passage percolation norm derived from (the law of) the edge weights. A dense subset of all norms in @math , and thus a large variety of shapes, arise from the class of weight distributions to which our proofs apply.
 This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice ℤd. We first introduce the model and give heuristic explanations of the long-time behavior of the solution, both in the annealed and the quenched setting for time-independent potentials. We thereby consider examples of potentials studied in the literature. In the particularly important case of an i.i.d. potential with double-exponential tails we formulate the asymptotic results in detail. Furthermore, we explain that, under mild regularity assumptions, there are only four different universality classes of asymptotic behaviors. Finally, we study the moment Lyapunov exponents for space-time homogeneous catalytic potentials generated by a Poisson field of random walks. For a > 0, let Wa(t) be the a-neighbourhood of standard Brownian motion in Rd starting at 0 and observed until time t. It is well-known that E|Wa(t)| ?at (t ? 8) for d = 3, with ?a the Newtonian capacity of the ball with radius a. We prove that @math and derive a variational representation for the rate function I?a . We show that the optimal strategy to realise the above moderate deviation is for Wa(t) to look like a Swiss cheese': Wa(t) has random holes whose sizes are of order 1 and whose density varies on scale t1 d. The optimal strategy is such that t-1 dWa(t) is delocalised in the limit as t ? 8. This is markedly different from the optimal strategy for large deviations |Wa(t)| = f(t) with f(t) = o(t), where Wa(t) is known to fill completely a ball of volume f(t) and nothing outside, so that Wa(t) has no holes and f(t)-1 dWa(t) is localised in the limit as t ? 8. We give a detailed analysis of the rate function I?a , in particular, its behaviour near the boundary points of (0, ?a) as well as certain monotonicity properties. It turns out that I?a has an infinite slope at ?a and, remarkably, for d = 5 is nonanalytic at some critical point in (0, ?a), above which it follows a pure power law. This crossover is associated with a collapse transition in the optimal strategy. We also derive the analogous moderate deviation result for d = 2. In this case E|Wa(t)| 2p t log t (t ? 8), and we prove that @math The rate function I2p has a finite slope at 2p. We study the boundary of the range for the simple random walk on Z d in the transient regime d ≥ 3. We show that sizes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the variance of order n log n in dimension three. We also establish a central limit theorem in dimension four and larger. We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the Swiss cheese picture of Bolthausen, van den Berg and den Hollander [BBH01]. 