We consider a random walk on the discrete cylinder $({ \mathbb{ Z}}/N{ \mathbb{ Z}})^d\times{ \mathbb{ Z}}$, $d\geq3$ with drift $N^{ -d\alpha}$ in the $\mathbb{ Z}$-direction and investigate the large $N$-behavior of the disconnection time $T^{ \mathrm{ disc}}_N$, 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 $T^{ \mathrm{ disc}}_N$ remains $N^{ 2d+o(1)}$, as in the unbiased case considered by Dembo and Sznitman, whereas for $\alpha<1$, the asymptotic behavior of $T^{ \mathrm{ disc}}_N$ becomes exponential in $N$.
Categories: math.PR