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

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)].


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)
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