Limit law for Brownian cover time of the two dimensional torus

Abstract

Let $\mathcal{C}_\varepsilon$ denote the time it takes a Brownian motion to come within distance $\varepsilon$ of every point in the two dimensional torus $\mathbb{T}^2$ of unit area. We prove that for an explicit non-random centering $m_\varepsilon$, the variables $\{\sqrt{\mathcal{C}_{\varepsilon}} - m_\varepsilon \}$ converge in distribution as $\varepsilon \to 0$, to a randomly shifted Gumbel law.