A Lower Bound on the Disconnection Time of a Discrete Cylinder

We study the asymptotic behavior for large N of the disconnection time T_N of simple random walk on a discrete cylinder with base a d-dimensional discrete torus of side-length N. When d is sufficiently large, we are able to substantially improve the lower bounds obtained by the authors in a previous article when d is bigger or equal to 2. We show here that the laws of N^(2d)/T_N are tight.