Toeplitz determinants with merging singularities

Abstract

We study asymptotic behavior for the determinants of $n\times n$ Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance $2t\ge 0$ 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 $t\to 0$ 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.