On the disconnection of a discrete cylinder by a biased random walk

We consider a random walk on the discrete cylinder $(\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^d\times \mathbb{Z}$, $d\ge 3$ with drift $N^{-d\alpha }$ in the $\mathbb{Z}$-direction and investigate the large $N$-behavior of the disconnection time $T_N^{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}$, defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent $\alpha$ is strictly greater than 1, the asymptotic behavior of $T_N^{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}$ remains $N^{2d+o(1)}$, as in the unbiased case considered by Dembo and Sznitman, whereas for $\alpha <1$, the asymptotic behavior of $T_N^{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}$ becomes exponential in $N$.