  We consider simple random walk on a discrete cylinder with base a large
d-dimensional torus of side-length N, when d is two or more. We develop a
stochastic domination control on the local picture left by the random walk in
boxes of side-length almost of order N, at certain random times comparable to
the square of the number of sites in the base. We show a domination control in
terms of the trace left in similar boxes by random interlacements in the
infinite (d+1)-dimensional cubic lattice at a suitably adjusted level. As an
application we derive a lower bound on the disconnection time of the discrete
cylinder, which as a by-product shows the tightness of the laws of the ratio of
the square of the number of sites in the base to the disconnection time. This
fact had previously only been established when d is at least 17, in arXiv:
math/0701414.
