The real layering field of Brownian loop soup and the Gaussian multiplicative chaos

TL;DR

This paper demonstrates that the layering field derived from Brownian loop soup converges to Gaussian multiplicative chaos after renormalization, using Wiener-Itô chaos expansion and analyzing conformal properties.

Contribution

It establishes the convergence of the Brownian loop soup layering field to Gaussian multiplicative chaos and computes its n-point functions with conformal covariance.

Findings

Layering field converges to Gaussian multiplicative chaos after renormalization.

Explicit calculation of n-point functions and their conformal covariance.

Behavior of the layering field near domain boundaries analyzed.

Abstract

We consider the random field defined by the layering numbers of the Brownian loop soup in a bounded simply connected domain in the complex plane. We call this the layering field and show that, after a suitable renormalization, it converges to the subcritical Gaussian multiplicative chaos. The main technique for our proof is the Wiener-It\^{o} chaos expansion. We also calculate the $n$-point functions of the layering field, show their conformal covariance and discuss their behavior near the boundary of the domain.