Delocalising the parabolic Anderson model through partial duplication of the potential

• 成果类型: Article
• 署名作者: Muirhead, Stephen; Pymar, Richard; Sidorova, Nadia
• 署名单位: University of Oxford; University of London; University College London; University of London; University College London; University of London; Birkbeck University London
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0798-5
• 发表日期: 2018
• 页码: 917-979
• 关键词: intermittency; localization

摘要

The parabolic Anderson model on 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.