Abstract:

The 2D Brownian loop soup is a Poisson point process of Brownian loops, with an intensity parameter theta>0. Sheffield and Werner showed that there is a phase transition for the clusters of the loop soup at theta=1/2. For theta less or equal to 1/2, the outer boundaries of clusters are distributed as Conformal Loop Ensembles CLE_\kappa. At the critical intensity theta=1/2, the clusters correspond to excursion sets of the 2D Gaussian free field, which comes from isomorphism theorems. This relation to the GFF implies in particular that the one arm probability in the Brownian loop soup theta=1/2 behaves like log to the power -1/2. In our recent work, we identify the behavior of the one arm probability as log to the power -(1-theta) for all the subcritical regime theta in (0,1/2). Our method relies on a dimension reduction. We further construct conformally invariant random fields related to these clusters of the Brownian loop soup, which are in a sense generalizations of the GFF.

https://arxiv.org/abs/2303.03782

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