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On large deviations for the range of a two-dimensional random walk

Serguei Popov, Quirin Vogel

Abstract

In this note, we compute the probability that a two-dimensional symmetric random walk visits more vertices than expected, for deviations on scales between the mean behavior and linear growth.

On large deviations for the range of a two-dimensional random walk

Abstract

In this note, we compute the probability that a two-dimensional symmetric random walk visits more vertices than expected, for deviations on scales between the mean behavior and linear growth.
Paper Structure (3 sections, 2 theorems, 39 equations, 1 figure)

This paper contains 3 sections, 2 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction and results
  2. Upper bound
  3. Lower bound

Key Result

Theorem 1

There exist $C_{\mathrm{up}}, C_{\mathrm{low}}>0$ such that for all $\theta_n\ge 1$ with $\theta_nr_n\le n$, we obtain that for all $n$

Figures (1)

  • Figure 1: Definition of the events $B_k$ and the strategy for the lower bound.

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 2.1
  • proof