### Phase transitions in the asymptotics for Toeplitz determinants

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.

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