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Tail Profile of Bulk Gaussian Multiplicative Chaos Measures I: Bulk/Boundary Quotients

Yichao Huang

TL;DR

This work analyzes the right tail problem for bulk Gaussian multiplicative chaos in the half-plane with a uniform boundary singularity, focusing on joint moments of bulk and boundary measures. By introducing a localization trick at the boundary and a non-convex variant of Kahane's inequality, the authors derive precise finiteness conditions for joint bulk/boundary quotients $\mathbb{E}[\mu^{\mathrm{H}}(Q)^{p}/\mu^{\partial}(I)^{q}]$, notably $p<\min(\tfrac{2}{\gamma^{2}}+\tfrac{q}{2}, \tfrac{4}{\gamma^{2}})$ for general $p,q\ge0$ and the sharper bound when $q=1$. They establish exact scaling relations, both in the bulk and in localized boundary situations, and develop an induction framework that connects $(p,q)$-moments to $(p,q+1)$-moments via a boundary localization approach. The results lay groundwork for the right-tail profile of the bulk measure and have potential implications for boundary Liouville conformal field theory, including the reflection coefficient and integrability questions. Overall, the paper provides robust, technically precise moment bounds and scaling identities that advance the understanding of bulk/boundary interactions in Gaussian multiplicative chaos near the boundary.

Abstract

This is the first part of a series of papers devoted to studying the right tail profile of a bulk Gaussian multiplicative chaos measure with uniform singularity on the boundary. We investigate the bulk/boundary quotients of Gaussian multiplicative chaos measures appearing in boundary Liouville conformal field theory, for which we establish preliminary joint moment bounds. These moment bounds will be a crucial ingredient in establishing the right tail profile of the bulk Gaussian multiplicative chaos measure in subsequent papers. The main idea is to implement the so-called localization trick at the boundary, and we also record a useful generalization of Kahane's convexity inequality, which is of independent interest. The study of the universal tail profiles of general bulk measures and bulk/boundary quotients as well as connections to integrability results of boundary Liouville conformal field theory will be pursued in subsequent papers.

Tail Profile of Bulk Gaussian Multiplicative Chaos Measures I: Bulk/Boundary Quotients

TL;DR

This work analyzes the right tail problem for bulk Gaussian multiplicative chaos in the half-plane with a uniform boundary singularity, focusing on joint moments of bulk and boundary measures. By introducing a localization trick at the boundary and a non-convex variant of Kahane's inequality, the authors derive precise finiteness conditions for joint bulk/boundary quotients , notably for general and the sharper bound when . They establish exact scaling relations, both in the bulk and in localized boundary situations, and develop an induction framework that connects -moments to -moments via a boundary localization approach. The results lay groundwork for the right-tail profile of the bulk measure and have potential implications for boundary Liouville conformal field theory, including the reflection coefficient and integrability questions. Overall, the paper provides robust, technically precise moment bounds and scaling identities that advance the understanding of bulk/boundary interactions in Gaussian multiplicative chaos near the boundary.

Abstract

This is the first part of a series of papers devoted to studying the right tail profile of a bulk Gaussian multiplicative chaos measure with uniform singularity on the boundary. We investigate the bulk/boundary quotients of Gaussian multiplicative chaos measures appearing in boundary Liouville conformal field theory, for which we establish preliminary joint moment bounds. These moment bounds will be a crucial ingredient in establishing the right tail profile of the bulk Gaussian multiplicative chaos measure in subsequent papers. The main idea is to implement the so-called localization trick at the boundary, and we also record a useful generalization of Kahane's convexity inequality, which is of independent interest. The study of the universal tail profiles of general bulk measures and bulk/boundary quotients as well as connections to integrability results of boundary Liouville conformal field theory will be pursued in subsequent papers.

Paper Structure

This paper contains 25 sections, 16 theorems, 106 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results of this paper
  3. Structure of this paper
  4. Mathematical background on Gaussian multiplicative chaos measures
  5. Gaussian multiplicative chaos
  6. Gaussian multiplicative chaos with boundary singularity
  7. Kahane's convexity inequality
  8. A non-convex variant of Kahane's inequality
  9. Gaussian coupling lemma
  10. Generalized Kahane's inequality for normalized exponentials of Gaussian fields
  11. Phenomenology and strategy of proofs
  12. Some observations on the bulk Gaussian multiplicative measure
  13. From $q$ to $q+1$
  14. Joint moment bounds and proofs of the main result
  15. Finiteness of quotients with positive distance
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose that $Q$ is a Carleson cube of the upper half-plane near the origin, say $[-r,r]\times[0,r]\subset\overline{\mathbb{H}}$ for some small $r>0$, and $I$ is its intersection with $\mathbb{R}$ (in this case $[-r,r]\subset\mathbb{R}$). With the notations in the previous section and $q=1$, the $(p if $p<\min(\frac{2}{\gamma^2}+\frac{1}{2},\frac{4}{\gamma^2})$. In particular, $p$ can be strictly

Figures (5)

  • Figure 1: Dividing a Carleson cube into infinitely many smaller cubes $(Q^{(n)}_i)_{n\geq 0, i=1\dots,2^{n}}$.
  • Figure 2: Tiling a Carleson cube by $\Gamma$-shaped regions (with respect to the bottom right end point).
  • Figure 3: Tiling a Carleson cube by $\Pi$-shaped regions (with respect to the middle point).
  • Figure 4: Superposing a $\Gamma$-tiling and a rescaled $\Pi$-tiling with respect to any point on the boundary (the $\Pi$-tiling starts with the blue regions and converges to the point $v$ at the boundry).
  • Figure 5: Covering a $\Gamma$-shape region between two dyadic scales by a Carleson cube and a region far from the boundary.

Theorems & Definitions (32)

  • Theorem 1: Finiteness of moments for bulk/boundary quotients, case $q=1$
  • Theorem 2: Finiteness of joint moments for bulk/boundary quotients
  • Remark 3
  • Lemma 4: Kahane's convexity inequality
  • Lemma 5
  • proof
  • Theorem 6: Non-convex Kahane's inequality for normalized exponentials of Gaussian fields
  • proof
  • Lemma 7: Decorrelating bulk/boundary quotients: an upper bound
  • proof
  • ...and 22 more