Thick points of random walk and the Gaussian free field

Abstract

We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of Dembo, Peres, Rosen and Zeitouni and compute the number of thick points of planar random walk, assuming that the increments are symmetric and have a finite moment of order two. The proof provides a streamlined argument based on the connection to the Gaussian free field and works in a very general setting including isoradial graphs. In higher dimensions, we study the scaling limit of the thick points. In particular, we show that the rescaled number of thick points converges to a nondegenerate random variable and that the centred maximum of the local times converges to a randomly shifted Gumbel distribution.

arXiv commentJournal version. 40 pages, 2 figures

References in corpus (10)

• The Laplacian and $\bar{\partial }$ operators on critical planar graphs
• Thick points of the Gaussian free field
• Extremes of the discrete two-dimensional Gaussian free field
• Planar Brownian motion and Gaussian multiplicative chaos
• On intermediate level sets of two-dimensional discrete Gaussian Free Field
• Characterisation of planar Brownian multiplicative chaos
• Exceptional points of two-dimensional random walks at multiples of the cover time
• Lectures on Isomorphism Theorems
• Maximal local time of a d-dimensional simple random walk on subsets
• Heavy points of a d-dimensional simple random walk

Cited by in corpus (11)

• Planar Brownian motion and Gaussian multiplicative chaos
• Characterisation of planar Brownian multiplicative chaos
• Critical Gaussian multiplicative chaos: a review
• Exceptional points of two-dimensional random walks at multiples of the cover time
• Critical Brownian multiplicative chaos
• Multiplicative chaos of the Brownian loop soup
• The geometry of random walk isomorphism theorems
• A limit law for the most favorite point of simple random walk on a regular tree
• Thick points of 4D critical branching Brownian motion
• Favorite sites for simple random walk in two and more dimensions
• A Ray-Knight theorem for $\nabla \varphi$ interface models and scaling limits