Characterisation of Planar Brownian Multiplicative Chaos

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.