Critical Brownian multiplicative chaos

Thu May 12th 2022, 3:15pm
Geology Corner 220
Antoine Jego, MSRI, Berkeley

In recent years, much effort has been put in establishing convergence of the (appropriately centered) maximum of logarithmically correlated fields. Famous examples where partial or complete solutions have been made include 1) branching random walk, 2) the Gaussian free field, 3) cover time of 2D random walk, and 4) thick points of 2D random walk, that is, points that have been visited unusually often by the trajectory. This talk is focused on the fourth example which is particularly challenging because of the lack of exact tree structure and Gaussianity of the model. Our aim is to construct a random variable that conjecturally describes the limiting distribution of the maximum. This random variable is described in terms of a critical multiplicative chaos measure / derivative martingale, providing the first example of a non-Gaussian critical multiplicative chaos measure. We will put our results in perspective and explain the similarities and crucial differences with Gaussian multiplicative chaos theory.