Localisation of the monkey walk with an "impurity" at the origin
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 ℤd 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 ξ0 when the walker sits at the origin and ξ>ξ0 anywhere else. We prove that, for all ξ0∈ (0,1), if ξ is sufficiently close to 1, then the monkey walk "localises", i.e. the position of the walker at time n, Wn, 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= ξ0, for which it is known that Wn/√(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.
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BibTeX
@misc{armfield2026localisation,
title = {{Localisation of the monkey walk with an "impurity" at the origin}},
author = {Wilfred Armfield and Cécile Mailler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21864},
eprint = {2610.21864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21864},
}
RIS
TY - UNPB
TI - Localisation of the monkey walk with an "impurity" at the origin
AU - Wilfred Armfield
AU - Cécile Mailler
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21864
UR - https://arxiv.org/abs/2610.21864
AB - 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 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∈ (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/√(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.
ER -