Probability seminar: Jonathon Peterson

Monday, 1 June 2026, 2pm to 3pm

Title: Limit Theorems for self-interacting random walks: a Ray-Knight approach

Abstract: A Ray-Knight theorem is a description of the local time profile of a stochastic process when stopped at some inverse local time. Since a Ray-Knight theorem contains a lot of information about the underlying process, and since a number of results have been obtained for self-interacting random walk models by proving Ray-Knight theorems for the walk, one naturally wonders if a Ray-Knight theorem can be used directly to deduce the scaling limit of the walk. Somewhat surprisingly, a recent result of myself with Kosygina and Mountford shows that this is not the case.
In this talk, I will show that while Ray-Knight theorems are not sufficient for proving scaling limits, one can obtain a functional limit for the walk through what we call joint Ray-Knight theorems. As an application of our main result we prove scaling limits for the “true” self-avoiding walk and the polynomially self-repelling motion. The “true” self-avoiding walk converges to a process called the “true” self-repelling motion, confirming a conjecture of Toth and Werner, while the scaling limit of the polynomially self-repelling random walk appears to be a new stochastic process. This is based on joint work with Elena Kosygina, Laure Mareche, and Tom Mountford.

Speaker(s): Jonathon Peterson (Purdue University)

Series: Probability seminar

Venue: Mathematical Institute - L5 - L5 Mathematical Institute Woodstock Road Oxford Oxfordshire OX2 6GG United Kingdom

Department: Statistics (Department)

Organiser: Christina Goldschmidt, James Martin, Julien Berestycki