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A Limit in Law for the Cover Time and Last Visited Vertex of Wired Planar Domains

Oren Louidor, Santiago Saglietti

Abstract

We derive a scaling limit in law for the cover time of a simple random walk on a lattice version of a scaled-up planar domain with wired boundary conditions. The limiting distribution is that of a Gumbel Random Variable shifted randomly by an independent quantity which is equal to the full mass of a variant of the critical Liouville Quantum Gravity Measure on the same domain. We also derive a limit in law for the scaled location of the vertex visited last by the walk. Here the limit turns out to be precisely the critical Liouville Measure, normalized by its total mass. Both limits hold jointly with the limiting joint law explicitly described. These results resolve well known open problems in the field, in the case of wired boundary conditions. The proof is based on comparison with the extremal landscape of the discrete Gaussian Free Field, and in particular a version there-of obtained by conditioning the average value of the field to be zero.

A Limit in Law for the Cover Time and Last Visited Vertex of Wired Planar Domains

Abstract

We derive a scaling limit in law for the cover time of a simple random walk on a lattice version of a scaled-up planar domain with wired boundary conditions. The limiting distribution is that of a Gumbel Random Variable shifted randomly by an independent quantity which is equal to the full mass of a variant of the critical Liouville Quantum Gravity Measure on the same domain. We also derive a limit in law for the scaled location of the vertex visited last by the walk. Here the limit turns out to be precisely the critical Liouville Measure, normalized by its total mass. Both limits hold jointly with the limiting joint law explicitly described. These results resolve well known open problems in the field, in the case of wired boundary conditions. The proof is based on comparison with the extremal landscape of the discrete Gaussian Free Field, and in particular a version there-of obtained by conditioning the average value of the field to be zero.
Paper Structure (42 sections, 56 theorems, 467 equations)

This paper contains 42 sections, 56 theorems, 467 equations.

Table of Contents

  1. Introduction and results
  2. Setup and main results
  3. Context and open problems
  4. Local time extreme order statistics, logarithmically correlated fields and Gaussian multiplicative chaos
  5. Historical account
  6. Open problems and conjectures
  7. Existence and identification of the limit: Top-level proofs of Theorems \ref{['t:1.1']}, \ref{['t:1.3i']}
  8. Notation
  9. Time parametrization and rescaling
  10. Separation into Phases A and B
  11. Phase A
  12. Phase B
  13. Proof of the main theorem
  14. Phase A: Top-level proof of Theorem \ref{['t:2.1']}
  15. Consequences of The Isomorphism
  16. ...and 27 more sections

Key Result

Theorem I

Let $\widecheck{{\mathbf t}}_N$ be defined via the relation: Then, there exist random variables $\widecheck{{\mathbf T}}_{\rm D}$ and $\widecheck{{\mathbf X}}_{\rm D}$, defined on the same probability space and take values in $\mathbb{R}$ and ${\rm D}$ respectively, such that

Theorems & Definitions (99)

  • Theorem I
  • Theorem II
  • Theorem III
  • Theorem IV
  • Theorem A: Phase A
  • Theorem B: Phase B
  • proof : Proof of Theorems \ref{['t:1.1']} and \ref{['t:1.3i']}
  • Theorem 3.1: Generalized second Ray-Knight Theorem
  • Theorem 3.2: Theorem 5.1 and Theorem 6.1 in Tightness
  • Theorem 3.3: Theorem 2.3 in Tightness
  • ...and 89 more