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.
