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Hölder type estimates for Gaussian multiplicative chaos

Yulai Huang

TL;DR

The paper analyzes the right tail behavior of balanced Gaussian multiplicative chaos (GMC) ratios Q_h(S) formed from log-correlated Gaussian fields h. It introduces a framework based on star-scale invariant kernels and a cascade mechanism to derive sharp tail estimates, proving a sub-exponential upper bound with exponent 4/(\alpha\gamma)-\delta and a matching lower bound under a decomposition h= X+Z with X star-scale invariant. Positive moments for balanced ratios are established, and a rigorous extension to tilted ratios L_{f,t} provides a robust recursive structure to control moments. These results advance the understanding of GMC ratio tails, connecting Hölder-type inequalities to the probabilistic structure of GMC and providing insights with potential applications to Liouville quantum gravity and derivative GMC phenomena.

Abstract

We investigate the right tail behavior of a certain class of GMC ratios, reminiscent of Hölder's inequality. We start with a heuristic argument to justify the optimal exponent in the tail estimate. Since Kahane's convexity inequality does not apply to GMC ratios, implementing the heuristic in the continuous setting is nontrivial from the viewpoint of GMC theory. We address the problem by enlarging the class of GMC ratios considered, and deduce the upper and lower bounds for the right tail of GMC ratios.

Hölder type estimates for Gaussian multiplicative chaos

TL;DR

The paper analyzes the right tail behavior of balanced Gaussian multiplicative chaos (GMC) ratios Q_h(S) formed from log-correlated Gaussian fields h. It introduces a framework based on star-scale invariant kernels and a cascade mechanism to derive sharp tail estimates, proving a sub-exponential upper bound with exponent 4/(\alpha\gamma)-\delta and a matching lower bound under a decomposition h= X+Z with X star-scale invariant. Positive moments for balanced ratios are established, and a rigorous extension to tilted ratios L_{f,t} provides a robust recursive structure to control moments. These results advance the understanding of GMC ratio tails, connecting Hölder-type inequalities to the probabilistic structure of GMC and providing insights with potential applications to Liouville quantum gravity and derivative GMC phenomena.

Abstract

We investigate the right tail behavior of a certain class of GMC ratios, reminiscent of Hölder's inequality. We start with a heuristic argument to justify the optimal exponent in the tail estimate. Since Kahane's convexity inequality does not apply to GMC ratios, implementing the heuristic in the continuous setting is nontrivial from the viewpoint of GMC theory. We address the problem by enlarging the class of GMC ratios considered, and deduce the upper and lower bounds for the right tail of GMC ratios.

Paper Structure

This paper contains 20 sections, 19 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Organization of the paper
  4. Acknowledgements
  5. Backgrounds and heuristics
  6. Star-scale invariant kernels
  7. Cascade relation
  8. Positive moments of balanced ratios
  9. Scaling relations for GMC ratios
  10. Proof of Proposition \ref{['moment_fini']}
  11. Proof of Theorem \ref{['moment_gen']}
  12. Recursive moment bounds for star-scale invariant kernels
  13. Tail estimates for balanced ratios
  14. Proof of Theorem \ref{['upper']}
  15. Proof of Theorem \ref{['lower']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any distinct $\alpha,\gamma\in(0,2)$ and $0<\delta<\frac{4}{\alpha\gamma}$, there exists some constants $C,c>0$ such that

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • ...and 23 more