2016/12/30 by Stephen Muirhead, Muirhead, Stephen, Richard Pymar +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR
paper · pdf · doi:10.48550/arxiv.1612.09583
34 pages; published version
openalex publication_date 2016/12/30 · arxiv created 2018/12/06 · arxiv updated 2018/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate a variant of the parabolic Anderson model, introduced in previous work, in which an i.i.d. potential is partially duplicated in a symmetric way about the origin, with each potential value duplicated independently with a certain probability. In previous work we established a phase transition for this model on the integers in the case of Pareto distributed potential with parameter α> 1 and fixed duplication probability p ∈ (0, 1): if α≥ 2 the model completely localises, whereas if α∈ (1, 2) the model may localise on two sites. In this paper we prove a new phase transition in the case that α≥ 2 is fixed but the duplication probability p(n) varies with the distance from the origin. We identify a critical scale p(n) → 1, depending on α, below which the model completely localises and above which the model localises on exactly two sites. We further establish the behaviour of the model in the critical regime.