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On the domination of a random walk on a discrete cylinder by random interlacements. (English) Zbl 1196.60170
Summary: We consider simple random walk on a discrete cylinder with base a large \(d\)-dimensional torus of side-length \(N\), when \(d\geq 2\). We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order \(N^{1-\varepsilon}\), with \(0<\varepsilon<1\), at certain random times comparable to \(N^{2d}\), in terms of the trace left in a similar box of \(\mathbb Z^{d+1}\) 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 \(d\geq 2\). This fact had previously only been established when \(d\) is at least 17, in [A. Dembo and A.-S. Sznitman, A lower bound on the disconnection time of a discrete cylinder. Basel: Birkhäuser. Progress in Probability 60, 211–227 (2008; Zbl 1173.82360)].

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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