Generalized intersection exponents and local cut points for three-dimensional Brownian loop soup

TL;DR

This paper introduces generalized intersection exponents for 3D Brownian loop soups, establishing their properties, relating them to local cut points' Hausdorff dimension, and analyzing their behavior at low intensities.

Contribution

It defines and proves properties of generalized intersection exponents for 3D Brownian loop soups, linking them to the geometry of local cut points.

Findings

Existence of generalized intersection exponents (GIE) for 3D Brownian loop soups.

GIE is continuous at zero intensity, matching classical exponents.

Set of local cut points has Hausdorff dimension > 1 for small intensities.

Abstract

We study generalized non-intersection probabilities for the three-dimensional Brownian loop soup at subcritical intensities. We establish the existence of generalized intersection exponents (GIE) and prove an up-to-constants estimate for these probabilities by means of a separation lemma tailored to this setting. We also relate the Hausdorff dimension of the set of local cut points of the three-dimensional Brownian loop soup to the GIE, and show that the GIE is continuous at intensity zero, where it reduces to the classical Brownian intersection exponent. In particular, this implies that, for sufficiently small intensity parameters, the set of local cut points has Hausdorff dimension strictly larger than $1$.