  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.
