The parabolic Anderson problem is the Cauchy problem for the heat equationWe consider independent and identically distributed potentials, such that the distribution function of ξ(z) converges polynomially at infinity. If u is initially localized in the origin, that is, if u(0, z) = ½0(z), we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential ξ. We consider the case when {ξ(z) : z ∈ Z d } is a collection of independent identically distributed random variables with Weibull distribution with parameter 0 < γ < 2, and we assume that the solution is initially localised in the origin. We prove that, as time goes to infinity, the solution completely localises at just one point with high probability, and we identify the asymptotic behaviour of the localisation site. We also show that the intervals between the times when the solution relocalises from one site to another increase linearly over time, a phenomenon known as ageing.

The parabolic Anderson model on Z d with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small values of the exponent we show that the model may delocalise. More precisely, we show that there is an event of non-negligible probability on which the solution has non-negligible mass on two sites.

The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model. MSc classification (2000): Primary 60K37 Secondary 82C441 Two recent papers, Dembo and Deuschel [DD07] and Aurzada and Doering [AD09], investigate weaker forms of ageing based on correlations. Both deal with a class of models which includes as a special case a parabolic Anderson model with time-variable potential and show absence of correlation-based ageing in this case. While this approach is probably the only way to deal rigorously with complicated models, it is not established that the effect picked up by these studies is actually really due to the existence or absence of ageing in our sense, or whether other moment effects are accountable.In the present work we show that the parabolic Anderson model exhibits ageing behaviour, at least if the underlying random potential is sufficiently heavy-tailed. As a lattice model with random disorder the parabolic Anderson model is a model of significant complexity, but its linearity and strong localization features make it considerably easier to study than, for example, the dynamics of most non-mean field spin glass models.Our work has led to three main results. The first one, Theorem 1.1, shows that the probability that during the time window [t, t + θt] the profiles of the solution of the parabolic Anderson problem remain within distance ε > 0 of each other converges to a constant I(θ), which is strictly between zero and one. This shows that ageing holds on a linear time scale. Our second main result, Theorem 1.3, is an almost sure ageing result. We define a function R(t) which characterizes the waiting time starting from time t until the profile changes again. We determine the precise almost sure upper envelope of R(t) in terms of an integral test. The third main result, Theorem 1.6, is a functional scaling limit theorem for the location of the peak, which determines the profile, and for the growth rate of the solution. We give the precise statements of the results in Section 1.2, and in Section 1.3 we provide a rough guide to the proofs. Statement of the main resultsThe parabolic Anderson model is given by the heat equation on the lattice Z d with a random po...

We consider a classical dilute particle system in a large box with pair- interaction given by a Lennard-Jones-type potential. The inverse temperature is picked proportionally to the logarithm of the particle density. We identify the free energy per particle in terms of a variational formula and show that this formula exhibits a cascade of phase transitions as the temperature parameter ranges from zero to infinity. Loosely speaking, the particle system separates into spatially distant components in such a way that within each phase all components are of the same size, which is the larger the lower the temperature. The main tool in our proof is a new large deviation principle for sparse point configurations