Electronic Journal of Probability

Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field

Artiom Fiodorov and Stephen Muirhead

Full-text: Open access

Abstract

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.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 58, 27 pp.

Dates
Accepted: 5 July 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065700

Digital Object Identifier
doi:10.1214/EJP.v19-3203

Mathematical Reviews number (MathSciNet)
MR3238778

Zentralblatt MATH identifier
1316.60109

Subjects
Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60F10: Large deviations 35P05: General topics in linear spectral theory

Keywords
Parabolic Anderson model Anderson Hamiltonian random Schrodinger operator localisation intermittency Weibull tail spectral gap

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Fiodorov, Artiom; Muirhead, Stephen. Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field. Electron. J. Probab. 19 (2014), paper no. 58, 27 pp. doi:10.1214/EJP.v19-3203. https://projecteuclid.org/euclid.ejp/1465065700


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