Characterisation of planar Brownian multiplicative chaos

Abstract

We characterise the multiplicative chaos measure $\mathcal{M}$ associated to planar Brownian motion introduced in [BBK94,AHS20,Jeg20a] by showing that it is the only random Borel measure satisfying a list of natural properties. These properties only serve to fix the average value of the measure and to express a spatial Markov property. As a consequence of our characterisation, we establish the scaling limit of the set of thick points of planar simple random walk, stopped at the first exit time of a domain, by showing the weak convergence towards $\mathcal{M}$ of the point measure associated to the thick points. In particular, we obtain the convergence of the appropriately normalised number of thick points of random walk to a nondegenerate random variable. The normalising constant is different from that of the Gaussian free field, as conjectured in [Jeg20b]. These results cover the entire subcritical regime. A key new idea for this characterisation is to introduce measures describing the intersection between different Brownian trajectories and how they interact to create thick points.

To appear in Comm. Math. Phys. 45 pages

References in corpus (1)

• Late points for random walks in two dimensions

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