Toeplitz determinants with merging singularities

We study asymptotic behavior for the determinants of n × n Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance 2 t ≥ 0 from each other on the unit circle. We obtain large n asymptotics which are uniform for 0 < t < t 0 , where t 0 is fixed. They describe the transition as t → 0 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.