  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 $\alpha > 1$ and fixed duplication
probability $p \in (0, 1)$: if $\alpha \ge 2$ the model completely localises,
whereas if $\alpha \in (1, 2)$ the model may localise on two sites. In this
paper we prove a new phase transition in the case that $\alpha \ge 2$ is fixed
but the duplication probability $p(n)$ varies with the distance from the
origin. We identify a critical scale $p(n) \to 1$, depending on $\alpha$, 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.
