Random walks on discrete cylinders with large bases and random interlacements
\textit{A.-S. Sznitman} [Probab. Theory Relat. Fields 145, No. 1--2, 143--174 (2009; Zbl 1172.60316)] has introduced the model of random interlacements on \(\mathbb{Z}^{d+1}, d>2\), and explored its relation with the microscopic structure left by simple random walk on an infinite discrete cylinder \((\mathbb{Z}/N\mathbb{Z})^d \times Z\) by times of order \(N^{2d}\) . That is generalised to random walk trajectories on a cylinder of the form \(G_N\times \mathbb{Z}\), where \(G_N\) is a large finite connected weighted graph, running to a time of order \(|G_N|^2\), Under suitable assumptions , the set of points not visited by the random walk, in a neighborhood od a point with the \(\mathbb{Z}\) component of order \(|G_N|\), converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the local structure of the graph. As examples of \(G_N\), Sierpinski gasket and \(d\)-ary tree are considered.
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