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

Sayantan Maitra

TL;DR

The paper analyzes the real-valued layering field arising from the Brownian loop soup in a bounded domain and proves that, after renormalization, it converges to the subcritical Gaussian multiplicative chaos (GMC) as $\lambda \to \infty$ and $\beta \to 0$ with $\lambda\beta^2\to\xi^2$. It builds two parallel objects—the Poisson layering field $V_{\lambda,\beta,D}$ and the Gaussian layering field $W_{\xi,D}$—and establishes their existence, correlation structure, and conformal covariance, using Wiener–Itô chaos expansions as a central tool. The main result is that the Poisson layering field converges in finite-dimensional distributions to the Gaussian layering field in the subcritical regime, thereby connecting loop-soup layering to GMC in the real-valued setting and enriching the correspondence previously known for the imaginary layering field. The work complements prior imaginary-layering results by providing a rigorous real-valued framework and highlights the role of renormalization and chaos expansions in bridging probabilistic loop-soup constructions with GMC theory under conformal settings.

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-Itô 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.

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

TL;DR

The paper analyzes the real-valued layering field arising from the Brownian loop soup in a bounded domain and proves that, after renormalization, it converges to the subcritical Gaussian multiplicative chaos (GMC) as and with . It builds two parallel objects—the Poisson layering field and the Gaussian layering field —and establishes their existence, correlation structure, and conformal covariance, using Wiener–Itô chaos expansions as a central tool. The main result is that the Poisson layering field converges in finite-dimensional distributions to the Gaussian layering field in the subcritical regime, thereby connecting loop-soup layering to GMC in the real-valued setting and enriching the correspondence previously known for the imaginary layering field. The work complements prior imaginary-layering results by providing a rigorous real-valued framework and highlights the role of renormalization and chaos expansions in bridging probabilistic loop-soup constructions with GMC theory under conformal settings.

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-Itô chaos expansion. We also calculate the -point functions of the layering field, show their conformal covariance and discuss their behavior near the boundary of the domain.
Paper Structure (23 sections, 21 theorems, 149 equations)

This paper contains 23 sections, 21 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and results
  3. The loop measure
  4. The loop soup and its layering field
  5. The Gaussian layering field and the multiplicative chaos
  6. Main results
  7. Existence of the Poisson and Gaussian layering fields
  8. Correlation functions
  9. Proof of Theorem \ref{['thm:main_1']}
  10. Proof of Theorem \ref{['thm:main_2']}
  11. Convergence of the Poisson layering field to the Gaussian layering field
  12. Chaos expansions
  13. Proof of Theorem \ref{['thm:main_3']}
  14. Proof of Proposition \ref{['prop:alpha_near_bdry']}
  15. Proofs from Section \ref{['sec:PLF_GLF_exist']}
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $z \in D$ and $0<\delta<R<\infty$ be such that $B(z, R) \subset D$. Then we have,

Theorems & Definitions (30)

  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 20 more