Emergence of a singularity for Toeplitz determinants and Painlevé V. (English) Zbl 1298.47039

Claeys, T.; Its, A.; Krasovsky, I.

Duke Math. J. 160, No. 2, 207-262 (2011).

Authors’ abstract: 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 Szegő’s strong limit theorem. If \(t=0\), the symbol possesses a Fisher-Hartwig singularity. Letting \(t\to 0\), 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 Painlevé V transcendent. A particular case of our result complements the classical description of T. T. Wu, B. M. McCoy, C. A. Tracy and E. Barouch of the behavior of a 2-spin correlation function for a large distance between spins in the two-dimensional Ising model as the phase transition occurs [“Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region”, Phys. Rev. B 13, 316–374 (1976)].

Reviewer: Edgar Pereira (Natal)

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
34M55	Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
15A15	Determinants, permanents, traces, other special matrix functions
34M50	Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
15B52	Random matrices (algebraic aspects)

Keywords:

asymptotic expansions; Toeplitz determinants

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