Planar Brownian motion and Gaussian multiplicative chaos

Abstract

We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereby considerably refining estimates of Dembo, Peres, Rosen and Zeitouni.

Final version. To appear in the Annals of Probability. 47 pages, 1 figure

References in corpus (4)

• Subcritical multiplicative chaos for regularized counting statistics from random matrix theory
• Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation
• Thick points of random walk and the Gaussian free field
• Lectures on Isomorphism Theorems

Cited by in corpus (9)

• Characterisation of planar Brownian multiplicative chaos
• Thick points of random walk and the Gaussian free field
• Exceptional points of two-dimensional random walks at multiples of the cover time
• Critical Brownian multiplicative chaos
• Exceptional points of discrete-time random walks in planar domains
• Multiplicative chaos of the Brownian loop soup
• Maxima of log-correlated fields: some recent developments
• A limit law for the most favorite point of simple random walk on a regular tree
• Convergence for Complex Gaussian Multiplicative Chaos on phase boundaries