I will talk about the asymptotic behavior for determinants of large Toeplitz matrices. Those asymptotics
depend on the smoothness properties of the underlying symbol of the Toeplitz matrices. For smooth symbols
that do not wind around the origin, Szego's strong limit theorem gives a description of the asymptotics
for the determinants. For symbols with a nite number of Fisher-Hartwig singularities (combining a jump
discontinuity with a root-type singularity), dierent asymptotics have been obtained. These results give
rise to two types of phase transitions: one occurs when a Szego symbol is deformed to a symbol with
one Fisher-Hartwig singularity, another one occurs when a symbol is deformed in such a way that two
Fisher-Hartwig singularities merge. I will describe the asymptotics for the Toeplitz determinants in those
phase transitions in terms of special smooth solutions to the fth Painleve equation. I will also discuss two
applications where these critical transitions are of interest, one of them has its origin in number theory, the
other is the two-dimensional Ising model.
The talk will be based on joint work with A. Its and I. Krasovsky.