# Localisation of a random walk in dimensions $d \ge 3$

#### Budapest - Vienna Probability Seminar

Date: Friday, March 6, 2020 14:00 - 14:50
Speaker: Nathanael Berestycki (University of Vienna)
Location: Rényi Institute, Budapest
Series: Mathematics and CS Seminar
Host: M. Beiglböck, N. Berestycki, L. Erdös, J. Maas, F. Toninelli

### Abstract:

We study a self-attractive random walk such that each trajectory of length $N$ is penalized by a factor proportional to $\exp(−|R_N |)$, where $R_N$ is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately $\rho_d N^{1/(d+2) }$, for some explicit constant $\rho_d >0$. This proves a conjecture of Bolthausen (1994) who obtained this result in the case d = 2. Joint work with Raphael Cerf (Paris).