Computing expected moments of the Rényi parking problem on the circle

01/28/2023

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by   Markus Hegland, et al.

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A highly accurate and efficient method to compute the expected values of the count and sum of the centre vectors of a random maximal sized collection of non-overlapping unit diameter disks touching a fixed unit-diameter disk is presented. This extends earlier work on Rényi's parking problem [Magyar Tud. Akad. Mat. Kutató Int. Közl. 3 (1-2), 1958, pp. 109-127]. Underlying the method is a splitting of the the problem conditional on the value of the first disk. This splitting is proven and then used to derive integral equations for the expectations. These equations take a lower block triangular form. They are solved using substitution and approximation of the integrals to very high accuracy using a high degree polynomial approximation within the blocks.