  Let $\mathcal{T}_n$ be the cover time of two-dimensional discrete torus
$\mathbb{Z}^2_n=\mathbb{Z}^2/n\mathbb{Z}^2$. We prove that
$\mathbb{P}[\mathcal{T}_n\leq \frac{4}{\pi}\gamma n^2\ln^2
n]=\exp(-n^{2(1-\sqrt{\gamma})+o(1)})$ for $\gamma\in (0,1)$. One of the main
methods used in the proofs is the decoupling of the walker's trace into
independent excursions by means of soft local times.
