Localisation of the monkey walk with an "impurity" at the origin

Source abstract

The monkey walk is a model of reinforced random walk introduced by Boyer and Solis-Salas as a model for the exploratory behaviour of capuchin monkeys. At each step of the walk, the walker flips a biased coin to determine whether to take a simple, symmetric random walk step on $\mathbb{Z}^d$ or to "relocate", i.e. choose a time uniformly at random in its past and jump to where it was at that time. More recently, Boyer, Falcón-Cortés, Giuggioli and Majumdar looked at a modification of the model, where the probability to relocate is $ξ_0$ when the walker sits at the origin and $ξ>ξ_0$ anywhere else. We prove that, for all $ξ_0\in (0,1)$, if $ξ$ is sufficiently close to 1, then the monkey walk "localises", i.e. the position of the walker at time $n$, $W_n$, converges in distribution without any renormalisation. (We in fact prove convergence of the occupation measure of the walker.) This is in contrast with the case when $ξ= ξ_0$, for which it is known that $W_n/\sqrt{\log n}$ converges in distribution to a standard Gaussian (we say that the walk "diffuses"). Our proof relies on seeing the monkey walk as an infinitely-many colour Pólya urn (a.k.a. measure-valued Pólya process - MVPP) and then prove convergence of this urn, which is not straightforward as theorems from the literature on MVPPs do not apply directly.