Computing expected moments of the R\'enyi parking problem on the circle

TL;DR

This paper introduces a highly accurate and efficient method to compute expected values related to the Re9nyi parking problem on a circle, extending previous work with new integral equations and polynomial approximations.

Contribution

The paper develops a novel splitting approach and integral equations to compute moments of the Re9nyi parking problem with high accuracy, advancing analytical methods in this area.

Findings

Derived integral equations for expectations

Achieved high-accuracy polynomial approximations

Extended previous Re9nyi parking problem analysis

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.