Emergence of a singularity for Toeplitz determinants and Painlevé V

We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter t. For t positive, the symbols are regular so that the determinants obey Szego's strong limit theorem. If t = 0, the symbol possesses a Fisher-Hartwig singularity. Letting t -> 0 we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painleve V transcendent. A particular case of our result complements the classical description of Wu, McCoy, Tracy, and Barouch of the behavior of a 2-spin correlation function for a large distance between spins in the two-dimensional lsing model as the phase transition occurs.