When does a random disk first overlap an earlier one? An expansion of the geometric birthday problem

The expected index of the first disk that overlaps an earlier one, when disks of diameter σ are placed independently and uniformly with centres in a disk of radius R, has the formal expansion √(π/(2q)) + 2/3 + √3/(4π) + X√q + c2q in the pair-overlap probability q, whose constant 2/3 + √3/(4π) replaces the 2/3 of the discrete birthday problem. It comes from a summation of the cumulant expansion whose coefficients we prove up to four disks; the five-disk coefficient is reduced to the fifth virial coefficient of hard disks. The probability that three disks are pairwise disjoint has a closed form in elementary functions and dilogarithms of σ/R for all σ < R. The wall corrections come from the classical formula for the area of translations of a summand of a disk, applied to the intersection of the disks of radius R containing a configuration. Spitzer's combinatorial lemma expresses the path terms of the four-disk wall correction through moments of a random walk and reduces its four-cycle term to a truncated moment; what is left is one constant built from four integrals. MSC2020: 60D05; 82B05; 60C05; 52A22 Files: the paper (PDF, 10 pages), its flattened LaTeX source (ZIP) and the scripts and data (ZIP). The value of the fifth virial coefficient used here is taken from the paper with DOI 10.5281/zenodo.23097001.