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Crossing exponent of the Brownian loop soup

Vortrag von Dr. Antoine Jego
Datum: 12.06.23   Zeit: 16.00 - 17.00   Raum: Y27H28

In this talk, I will present a paper where we study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity θ ∈ (0, 1/2]. We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii r and rs as r → 0 (s > 1 fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius r decays like | log r|−1+θ+o(1) as r → 0. Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of logα-capacity.

This paper reveals a connection between the 1D and 2D Brownian loop soups. This connection in turn implies the existence of a second critical intensity θ = 1 that describes a phase transition in the percolative behaviour of large loops on a logarithmic scale targeting an interior point of the domain.

Based on a joint work with Titus Lupu and Wei Qian.