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Toeplitz determinants with merging singularities. (English) Zbl 1333.15018

The authors study the asymptotic behavior for the determinants of \(n\times n\) Toeplitz matrices corresponding to symbols with two Fisher-Hartwig singularities at the distance \(2t\geq 0\) from each other on the unit circle. They obtain large \(n\) asymptotics which are uniform for \(0<t<t_0\), where \(t_0\) is fixed. These describe the transition as \(t\to 0\) between the asymptotic regimes of two singularities and one singularity. The asymptotics involve a particular solution to the Painlevé V equation. They also obtain small and large argument expansions of this solution. As applications they prove a conjecture of F. Dyson [“Toeplitz determinants and Coulomb gases”, lecture held at Eastern theoretical physics conference, Chapel Hill, North Carolina, 1963] on the largest occupation number in the ground state of a one-dimensional Bose gas, and a conjecture of Y. V. Fyodorov and J. P. Keating [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 372, No. 2007, Article ID 20120503, 32 p. (2014; Zbl 1330.82028)] on the second moment of powers of the characteristic polynomials of random matrices.

MSC:

15B05 Toeplitz, Cauchy, and related matrices
15A15 Determinants, permanents, traces, other special matrix functions
33E17 Painlevé-type functions
35Q15 Riemann-Hilbert problems in context of PDEs
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
15B52 Random matrices (algebraic aspects)

Citations:

Zbl 1330.82028
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References:

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