Crossing exponent of the Brownian loop soup
Vortrag von Dr. Antoine Jego
Datum: 20.11.24 Zeit: 17.15 - 18.45 Raum: ETH HG G 43
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.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.