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Capacity of the range of random walk: The law of the iterated logarithm

Amir Dembo, Izumi Okada

Abstract

We establish both the $\limsup$ and the $\liminf$ law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension $d\ge 3$. While for $d \ge 4$, the order of growth in $n$ of such LIL at dimension $d$ matches that for the volume of the random walk range in dimension $d-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $d=3$. We further establish such LIL for the Brownian capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a $4$-dimensional simple random walk.

Capacity of the range of random walk: The law of the iterated logarithm

Abstract

We establish both the and the law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension . While for , the order of growth in of such LIL at dimension matches that for the volume of the random walk range in dimension , somewhat surprisingly this correspondence breaks down for the capacity of the range at . We further establish such LIL for the Brownian capacity of a -dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a -dimensional simple random walk.
Paper Structure (13 sections, 22 theorems, 266 equations)

This paper contains 13 sections, 22 theorems, 266 equations.

Table of Contents

  1. Introduction and Main Results
  2. Capacity geometry and Green's function
  3. lil for srw on ${\mathbb Z}^3$: Proof of Theorem \ref{['m1']}
  4. The lower bound in the limsup-lil
  5. The upper bound in the limsup-lil
  6. The upper bound in the liminf-lil
  7. The lower bound of the liminf-lil
  8. lil for srw on ${\mathbb Z}^4$: Proof of Theorem \ref{['m2']}
  9. The lower bound in the limsup-lil
  10. The upper bound in the limsup-lil
  11. Non-random and positive liminf-lil
  12. Finiteness of the liminf-lil
  13. lil for srw on ${\mathbb Z}^d$, $d \ge 5$: Proof of Theorem \ref{['m3']}

Key Result

Theorem 1.1

For $d=3$, almost surely, where

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • Remark
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 1.7
  • Lemma 2.1
  • Remark 2.2
  • ...and 37 more