On the domination of random walk on a discrete cylinder by random interlacements

We consider simple random walk on a discrete cylinder with base a large d-dimensional torus of side-length N, when d >= 2. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length of order N(1-epsilon), with 0 < epsilon < 1, at certain random times comparable to N(2d), in terms of the trace left in a similar box of Z(d+1) by random interlacements at a suitably adjusted level. As an application we derive a lower bound on the disconnection time T(N) of the discrete cylinder, which as a by-product shows the tightness of the laws of N(2d)/T(N), for all d >= 2. This fact had previously only been established when d >= 17, in [3].