A new phase transition in the parabolic Anderson model with partially duplicated potential.

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. [ABSTRACT FROM AUTHOR]