We study asymptotic behavior for the determinants of Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance from each other on the unit circle. We obtain large asymptotics which are uniform for , where is fixed. They describe the transition as between the asymptotic regimes of two singularities and one singularity. The asymptotics involve a particular solution to the Painlevé V equation. We obtain small and large argument expansions of this solution. As applications of our results, we prove a conjecture of Dyson on the largest occupation number in the ground state of a one-dimensional Bose gas, and a conjecture of Fyodorov and Keating on the second moment of powers of the characteristic polynomials of random matrices.
T. Claeys. I. Krasovsky. "Toeplitz determinants with merging singularities." Duke Math. J. 164 (15) 2897 - 2987, 1 December 2015. https://doi.org/10.1215/00127094-3164897