# Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos

## TL;DR

This paper constructs and analyzes a new class of conformally covariant random fields derived from Brownian loop soups, connecting them to imaginary Gaussian multiplicative chaos through a renormalization and limit process.

## Contribution

It introduces a novel approach to defining vertex-like fields from Brownian loop soups and establishes their convergence to a conformally covariant field related to imaginary Gaussian multiplicative chaos.

## Key findings

- Fields are well-defined in Sobolev spaces after renormalization.
- Vertex-like fields converge to a conformally covariant field as loop intensity increases.
- Explicit Wiener-Itô chaos expansion is used to analyze the fields.

## Abstract

We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. As smaller loops are included in the count, that sum diverges logarithmically with the diameter of the loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener-It\^{o} chaos expansion of linear functionals of vertex-like fields.