Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field. (English) Zbl 1316.60109

Fiodorov, Artiom; Muirhead, Stephen

Electron. J. Probab. 19, Paper No. 58, 27 p. (2014).

Summary: We consider the parabolic Anderson model with Weibull potential field, for all values of the Weibull parameter. We prove that the solution is eventually localised at a single site with overwhelming probability (complete localisation) and, moreover, that the solution has exponential shape around the localisation site. We determine the localisation site explicitly, and derive limit formulae for its distance, the profile of the nearby potential field and its ageing behaviour. We also prove that the localisation site is determined locally, that is, by maximising a certain time-dependent functional that depends only on: (i) the value of the potential field in a neighbourhood of fixed radius around a site; and (ii) the distance of that site to the origin. Our results extend the class of potential field distributions for which the parabolic Anderson model is known to completely localise; previously, this had only been established in the case where the potential field distribution has sub-Gaussian tail decay, corresponding to a Weibull parameter less than two.

MSC:

60H25	Random operators and equations (aspects of stochastic analysis)
60F10	Large deviations
35P05	General topics in linear spectral theory for PDEs
82C44	Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics

Keywords:

parabolic Anderson model; Anderson Hamiltonian; random Schrödinger operator; localisation; intermittency; Weibull tail; spectral gap