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Extremal properties of the random walk local time

Marek Biskup

TL;DR

This work develops a detailed framework for extremal phenomena of two-dimensional random walk local time by connecting it to the Gaussian Free Field via precise Green function asymptotics and the Dynkin/DK isomorphism. It constructs and analyzes random measures Z^D_λ that describe the limiting distribution of thick points of the DGFF and shows that the local-time zero-set at cover-time scales converges to the corresponding thick-point measure Z^D_{\sqrt{\theta}}. The results establish a deep link between local-time extremes, Gaussian multiplicative chaos, and Liouville quantum gravity, providing both exact limit theorems and a robust toolkit (Gibbs-Markov structure, Kac moments, Ray-Knight theorems) for studying extremal processes in logarithmically correlated systems. The findings have broad implications for understanding universality in logarithmically correlated fields and inform ongoing research on extremal processes, cover times, and their continuous counterparts. The work also outlines important open problems and conjectures for the natural time parametrization and for the maximal and cover-time statistics in varying geometries.

Abstract

These are expanded lecture notes for a minicourse taught at the "School on disordered media" at the Alfred Renyi institute in Budapest, January 2025.

Extremal properties of the random walk local time

TL;DR

This work develops a detailed framework for extremal phenomena of two-dimensional random walk local time by connecting it to the Gaussian Free Field via precise Green function asymptotics and the Dynkin/DK isomorphism. It constructs and analyzes random measures Z^D_λ that describe the limiting distribution of thick points of the DGFF and shows that the local-time zero-set at cover-time scales converges to the corresponding thick-point measure Z^D_{\sqrt{\theta}}. The results establish a deep link between local-time extremes, Gaussian multiplicative chaos, and Liouville quantum gravity, providing both exact limit theorems and a robust toolkit (Gibbs-Markov structure, Kac moments, Ray-Knight theorems) for studying extremal processes in logarithmically correlated systems. The findings have broad implications for understanding universality in logarithmically correlated fields and inform ongoing research on extremal processes, cover times, and their continuous counterparts. The work also outlines important open problems and conjectures for the natural time parametrization and for the maximal and cover-time statistics in varying geometries.

Abstract

These are expanded lecture notes for a minicourse taught at the "School on disordered media" at the Alfred Renyi institute in Budapest, January 2025.

Paper Structure

This paper contains 26 sections, 36 theorems, 241 equations, 10 figures.

Table of Contents

  1. Introduction and main results
  2. Random walk local time
  3. The case for $d=2$
  4. Link to Gaussian Free Field
  5. Main result on DGFF
  6. Main result on local time
  7. Green function asymptotic and connection to DGFF
  8. Asymptotic of the Green function
  9. Proof of Theorem \ref{['thm-2.1']}
  10. Connection between the local time and the DGFF
  11. Kac moment formula
  12. Proof of the Second Ray-Knight Theorem
  13. Thick points of the DGFF
  14. Gibbs-Markov property of DGFF
  15. Subsequential limits
  16. ...and 11 more sections

Key Result

Theorem 1.5

Let $\{D_N\}_{N\ge1}$ be admissible approximations of an admissible domain $D\subseteq\mathbb R^2$. There exists a family of a.s-finite random Borel measures $\{Z^D_\lambda\colon \lambda\in(0,1)\}$ on $D$ such that for any positive sequence $\{a_N\}_{n\ge1}$ with and with we have where $\alpha:=2/\sqrt g$ and the weak convergence is relative to vague topology on $\overline D\times\mathbb R$. Mo

Figures (10)

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  • ...and 5 more figures

Theorems & Definitions (42)

  • Definition 1.2
  • Definition 1.3: Admissible domain
  • Definition 1.4: Admissible approximations
  • Theorem 1.5: B.-Louidor BL4
  • Corollary 1.6
  • Theorem 1.7: Abe-B. AB
  • Corollary 1.8
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 32 more