Given a symmetric random walk in \({\mathbb Z}^2\) with finite second moments, let \(R_n\) be the range of the random walk up to time \(n\). The authors study moderate deviations for \(R_n -{\mathbb E}R_n\) and \({\mathbb E}R_n -R_n\). They also derive the corresponding laws of the iterated logarithm.

Table of Contents

• Introduction
• History
• Overview
• Preliminaries
• Moments of the range
• Moderate deviations for \(R_n-{\mathbb E}R_n\)
• Moderate deviations for \({\mathbb E}R_n -R_n\)
• Exponential asymptotics for the smoothed range
• Exponential approximation
• Laws of the iterated logarithm
• Bibliography