Computing expected moments of the R\'enyi parking problem on the circle
Abstract
A highly accurate and efficient method to compute the expected values of the count, sum, and squared norm of the 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 R\'enyi's parking problem [Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl. 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 polynomial approximation within the blocks.
Keywords
Cite
@article{arxiv.2301.12115,
title = {Computing expected moments of the R\'enyi parking problem on the circle},
author = {Markus Hegland and Conrad J. Burden and Zbigniew Stachurski},
journal= {arXiv preprint arXiv:2301.12115},
year = {2023}
}