Intensity doubling for Brownian loop-soups in high dimensions

TL;DR

This paper demonstrates that in high-dimensional Brownian loop-soups, large cycles can be viewed as originating from two independent sources, leading to a doubling of the critical intensity in the scaling limit.

Contribution

It proves that large cycles in high-dimensional Brownian loop-soups can be decomposed into two independent families, resulting in a doubled critical intensity in the scaling limit.

Findings

Large cycles are composed of two asymptotically independent families.

Large cycles formed by small loops resemble large Brownian loops.

The scaling limit exhibits a Brownian loop-soup with twice the critical intensity.

Abstract

We derive an intensity doubling feature of critical Brownian loop-soups on the cable-graphs of ${\mathbb Z}^d$ for $d \ge 7$ that can be described as follows: In the box $[-N, N]^d$ (and with a probability that goes to $1$ as $N$ goes to infinity), the set of all clusters of Brownian loops that do contain proper self-avoiding cycles of diameter comparable to $N$ can be decomposed into two identically distributed families: (a) The collection of clusters that do contain a large Brownian loop from the loop-soup (and therefore do automatically contain such a large cycle) (b) The collection of clusters that contain no macroscopic loop from the loop-soup (more specifically, no loop of diameter greater than $N^{\beta}$ when $\beta > 4/ (d-2)$ is fixed) but nevertheless contain a large cycle. In particular, due to the fact that these two families are asymptotically identically distributed, large cycles formed in case (b) by chains of small Brownian loops (i.e., all of diameter much smaller than $N$) will look like large Brownian loops themselves, and form a second independent "ghost" critical loop-soup in the scaling limit. Reformulated in terms of the Gaussian free field on such cable-graphs, this shows that large cycles in the collection of its sign clusters will converge in the scaling limit to a Brownian loop-soup with twice the usual critical intensity. This result had been conjectured by the first author in arXiv:2209.07901 [math.PR] ; our proof builds heavily on the second author's switching property for such loop-soups from arXiv:2502.06754 [math.PR] .