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July 2014 On the range of a random walk in a torus and random interlacements
Eviatar B. Procaccia, Eric Shellef
Ann. Probab. 42(4): 1590-1634 (July 2014). DOI: 10.1214/14-AOP924


Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices visited by the walk. Distance and mixing bounds for the typical range are proven that are a $k$-iterated log factor from those on the full torus for arbitrary $k$. The proof uses hierarchical renormalization and techniques that can possibly be applied to other random processes in the Euclidean lattice. We use the same technique to bound the heat kernel of a random walk on random interlacements.


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Eviatar B. Procaccia. Eric Shellef. "On the range of a random walk in a torus and random interlacements." Ann. Probab. 42 (4) 1590 - 1634, July 2014.


Published: July 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1305.60106
MathSciNet: MR3262486
Digital Object Identifier: 10.1214/14-AOP924

Primary: 60K35
Secondary: 60K37

Keywords: Mixing , Random interlacements , Random walk

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 4 • July 2014
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