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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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