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Exceptional points of discrete-time random walks in planar domains

Yoshihiro Abe, Marek Biskup, Sangchul Lee

Abstract

Given a sequence of lattice approximations $D_N\subset\mathbb Z^2$ of a bounded continuum domain $D\subset\mathbb R^2$ with the vertices outside $D_N$ fused together into one boundary vertex $\varrho$, we consider discrete-time simple random walks in $D_N\cup\{\varrho\}$ run for a time proportional to the expected cover time and describe the scaling limit of the exceptional level sets of the thick, thin, light and avoided points. We show that these are distributed, up a spatially-dependent log-normal factor, as the zero-average Liouville Quantum Gravity measures in $D$. The limit law of the local time configuration at, and nearby, the exceptional points is determined as well. The results extend earlier work by the first two authors who analyzed the continuous-time problem in the parametrization by the local time at $\varrho$. A novel uniqueness result concerning divisible random measures and, in particular, Gaussian Multiplicative Chaos, is derived as part of the proofs.

Exceptional points of discrete-time random walks in planar domains

Abstract

Given a sequence of lattice approximations of a bounded continuum domain with the vertices outside fused together into one boundary vertex , we consider discrete-time simple random walks in run for a time proportional to the expected cover time and describe the scaling limit of the exceptional level sets of the thick, thin, light and avoided points. We show that these are distributed, up a spatially-dependent log-normal factor, as the zero-average Liouville Quantum Gravity measures in . The limit law of the local time configuration at, and nearby, the exceptional points is determined as well. The results extend earlier work by the first two authors who analyzed the continuous-time problem in the parametrization by the local time at . A novel uniqueness result concerning divisible random measures and, in particular, Gaussian Multiplicative Chaos, is derived as part of the proofs.

Paper Structure

This paper contains 18 sections, 44 theorems, 332 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Setting the scales
  4. Level sets of zero-average DGFF
  5. Exceptional local-time sets
  6. Local structure
  7. Remarks
  8. Outline
  9. Zero average DGFF level sets
  10. Augmented boundary vertex measures
  11. Fixed total time
  12. Time conversions
  13. Continuous-time exceptional level sets
  14. Arbitrary starting points
  15. Discrete time conclusions
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $\{t_N\}_{N\ge1}$ be a positive sequence such that, for some $\theta>0$, Then for any choices of $x_N\in D_N$, the following limits hold in $P^{x_N}$-probability: and

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3: Thick points
  • Theorem 2.4: Thin points
  • Theorem 2.5: Light points
  • Theorem 2.6: Avoided points
  • Theorem 2.7: Local structure of thick/thin points
  • Theorem 2.8: Local structure of avoided points
  • ...and 36 more