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Characterisation of planar Brownian multiplicative chaos

Antoine Jego

TL;DR

The paper characterises the planar Brownian multiplicative chaos via a natural set of averaging, spatial Markov, independence, and non-atomicity properties, proving uniqueness of the chaos law. It then leverages this characterisation to show that thick points of planar random walk stopped at a domain exit converge, in a precise weak sense, to the Brownian chaos, with a nondegenerate limiting mass and a normalization differing from the Gaussian free field. A key innovation is the multipoint chaos and its intersection measures, which describe how multiple Brownian trajectories interact to create thick points and admit a detailed decomposition into independent path components. The results establish the full subcritical scaling limit for thick points and connect to broader structures such as Liouville-type measures and loop-soup frameworks, enabling extensions to joint convergence of measures and trajectories.

Abstract

We characterise the multiplicative chaos measure $\mathcal{M}$ associated to planar Brownian motion introduced in [BBK94,AHS20,Jeg20a] by showing that it is the only random Borel measure satisfying a list of natural properties. These properties only serve to fix the average value of the measure and to express a spatial Markov property. As a consequence of our characterisation, we establish the scaling limit of the set of thick points of planar simple random walk, stopped at the first exit time of a domain, by showing the weak convergence towards $\mathcal{M}$ of the point measure associated to the thick points. In particular, we obtain the convergence of the appropriately normalised number of thick points of random walk to a nondegenerate random variable. The normalising constant is different from that of the Gaussian free field, as conjectured in [Jeg20b]. These results cover the entire subcritical regime. A key new idea for this characterisation is to introduce measures describing the intersection between different Brownian trajectories and how they interact to create thick points.

Characterisation of planar Brownian multiplicative chaos

TL;DR

The paper characterises the planar Brownian multiplicative chaos via a natural set of averaging, spatial Markov, independence, and non-atomicity properties, proving uniqueness of the chaos law. It then leverages this characterisation to show that thick points of planar random walk stopped at a domain exit converge, in a precise weak sense, to the Brownian chaos, with a nondegenerate limiting mass and a normalization differing from the Gaussian free field. A key innovation is the multipoint chaos and its intersection measures, which describe how multiple Brownian trajectories interact to create thick points and admit a detailed decomposition into independent path components. The results establish the full subcritical scaling limit for thick points and connect to broader structures such as Liouville-type measures and loop-soup frameworks, enabling extensions to joint convergence of measures and trajectories.

Abstract

We characterise the multiplicative chaos measure associated to planar Brownian motion introduced in [BBK94,AHS20,Jeg20a] by showing that it is the only random Borel measure satisfying a list of natural properties. These properties only serve to fix the average value of the measure and to express a spatial Markov property. As a consequence of our characterisation, we establish the scaling limit of the set of thick points of planar simple random walk, stopped at the first exit time of a domain, by showing the weak convergence towards of the point measure associated to the thick points. In particular, we obtain the convergence of the appropriately normalised number of thick points of random walk to a nondegenerate random variable. The normalising constant is different from that of the Gaussian free field, as conjectured in [Jeg20b]. These results cover the entire subcritical regime. A key new idea for this characterisation is to introduce measures describing the intersection between different Brownian trajectories and how they interact to create thick points.

Paper Structure

This paper contains 14 sections, 24 theorems, 218 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Scaling limit of thick points of planar random walk
  3. Brownian multiplicative chaos: background and extension
  4. Characterisation of Brownian multiplicative chaos
  5. Further results on multipoint Brownian multiplicative chaos
  6. Outline of proofs
  7. Some notations
  8. Characterisation: proof of Theorem \ref{['th:charac']}
  9. Application to random walk: proof of Theorem \ref{['th:convergence']}
  10. Tightness and first moment estimates
  11. Study of the subsequential limits
  12. Uniform integrability: proof of Proposition \ref{['prop:uniform_integrability']}
  13. Joint convergence of measures and trajectories
  14. Multipoint Brownian multiplicative chaos

Key Result

Theorem 1.1

For all $a \in (0,2)$, the sequence $\mu^{U,a}_{x_0;N}, N \geq 1,$ (resp. $\mu^{U,a}_{x_0,z;N}, N \geq 1$) converges weakly relatively to the topology of weak convergence (resp. vague convergence) on $U$. Moreover, the limiting measure has the same distribution as $e^{c_0a/g} \mathcal{M}^{U,a}_{x_0}

Figures (1)

  • Figure 1.1: Representation of the successive hitting points of $2^{-p} \mathbb{Z} \times \mathbb{R}$ by a Brownian motion sampled according to $\mathbb{P}_{x_0,z}^D$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 1.7: Disintegration
  • Proposition 1.8
  • proof : Proof of Theorem \ref{['th:charac']}, existence
  • proof : Proof of Theorem \ref{['th:charac']}, uniqueness
  • ...and 35 more