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A two cities theorem for the parabolic Anderson model. (English) Zbl 1183.60024

The authors study the parabolic Anderson model with potential distributions that do not have any finite exponential moment. Their main result is that, in the case of potentials with polynomial tails, almost surely at all large times there are at most two relevant islands, each of which consists of a single site. This means that the proportion of the total mass \(U(t)=\sum_{z\in Z^d}u(t,z)\), \(t>0\), is asymptotically concentrated in just two time-dependent lattice points. It is also proved that, with high probability, the total mass \(U(t)\) is concentrated in a single lattice point.
The main result is the so-called “two cities theorem”: at a typical large time, the mass (a population), inhabits one site (a city). At some rare times, a better site has been found and the entire population moves to the new site. At the transition times part of the population still live in the old city and another part has already moved to the new one. The authors also study the asymptotic locations of the points where the mass concentrates in terms of a weak limit theorem with an explicit limiting density.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60F10 Large deviations
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References:

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