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On the disconnection of a discrete cylinder by a biased random walk. (English) Zbl 1148.60028
The author considers a random walk \((X_n)_{n\geq0}\) with drift \(N^{-d\alpha}\), \(d\geq 3\), \(\alpha>0\), in the \(\mathbb Z\)-direction on the discrete cylinder \(\mathbb T_N^d\times \mathbb Z\), where \(\mathbb T_N^d\) denotes the \(d\)-dimensional integer torus \(({\mathbb Z}/N{\mathbb Z})^d\). The central object of interest is the disconnection time \(T_N\) defined as the first time when a particle moving randomly according to \((X_n)_{n\geq0}\), and damaging every visited point, breaks the cylinder into two infinite components.
This setting is a version of a model initiated by A. Dembo and A.-S. Sznitman [Probab. Theory Relat. Fields 136, 321–340 (2006; Zbl 1105.60029)]. The main result is that, as \(N\to \infty\), the asymptotic behaviour of \(T_N\) remains \(N^{2d+o(1)}\) (as in the unbiased case considered by Dembo and Sznitman) for \(\alpha>1\), whereas the asymptotic behaviour of \(T_N\) becomes exponential in \(N\) for \(\alpha<1\). Nothing is said about the case \(\alpha=1\).

MSC:
60G50 Sums of independent random variables; random walks
60D05 Geometric probability and stochastic geometry
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