Abstract
AbstractWe characterise the multiplicative chaos measure $${\mathcal {M}}$$
M
associated to planar Brownian motion introduced in Bass et al. (Ann Probab 22(2):566–625, 1994), Aïdékon et al. (Ann. Probab. 48(4), 1785–1825, 2020) and Jego (Ann Probab 48(4):1597–1643, 2020) 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 $${\mathcal {M}}$$
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 Jego (Electron J Probab 25:39, 2020). These results cover the entire subcritical regime. A key new idea for this characterisation is to introduce measures describing the intersection between different independent Brownian trajectories and how they interact to create thick points.
Funder
osterreichischen Akademie der Wissenschaften
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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