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On the nature of the Swiss cheese in dimension 3. (English) Zbl 1434.60081
Summary: We study scenarii linked with the Swiss cheese picture in dimension 3 obtained when two random walks are forced to meet often, or when one random walk is forced to squeeze its range. In the case of two random walks, we show that they most likely meet in a region of optimal density. In the case of one random walk, we show that a small range is reached by a strategy uniform in time. Both results rely on an original inequality estimating the cost of visiting sparse sites, and in the case of one random walk on the precise large deviation principle of M. van den Berg et al. [Ann. Math. (2) 153, No. 2, 355–406 (2001; Zbl 1004.60021)], including their sharp estimates of the rate functions in the neighborhood of the origin.

60F10 Large deviations
60G50 Sums of independent random variables; random walks
Full Text: DOI Euclid
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