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.


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)


Zbl 1330.82028
Full Text: DOI arXiv Euclid


[1] J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations , J. Amer. Math. Soc. 12 (1999), 1119-1178. · Zbl 0932.05001
[2] E. Basor, Asymptotic formulas for Toeplitz determinants , Trans. Amer. Math. Soc. 239 (1978), 33-65. · Zbl 0409.47018
[3] E. Basor, A localization theorem for Toeplitz determinants , Indiana Univ. Math. J. 28 (1979), 975-983. · Zbl 0396.47018
[4] A. Böttcher and B. Silbermann, Toeplitz operators and determinants generated by symbols with one Fisher-Hartwig singularity , Math. Nachr. 127 (1986), 95-123. · Zbl 0613.47024
[5] T. Claeys, A. Its, and I. Krasovsky, Emergence of a singularity for Toeplitz determinants and Painlevé V , Duke Math. J. 160 (2011), 207-262. · Zbl 1298.47039
[6] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , Courant Lect. Notes Math. 3 , Amer. Math. Soc., Providence, 1998. · Zbl 0997.47033
[7] P. Deift, A. Its, and I. Krasovsky, Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities , Ann. of Math. (2) 174 (2011), 1243-1299. · Zbl 1232.15006
[8] P. Deift, A. Its, and I. Krasovsky, Eigenvalues of Toeplitz matrices in the bulk of the spectrum , Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), 437-461. · Zbl 1292.15029
[9] P. Deift, A. Its, and I. Krasovsky, Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: Some history and some recent results , Comm. Pure Appl. Math. 66 (2013), 1360-1438. · Zbl 1292.47016
[10] P. Deift, A. Its, and I. Krasovsky, “ On the asymptotics of a Toeplitz determinant with singularities ” in Random Matrix Theory, Interacting Particle Systems, and Integrable Systems , Cambridge Univ. Press, New York, 2014, 93-146. · Zbl 1326.35218
[11] F. Dyson, Toeplitz determinants and Coulomb gases , conference lecture at Eastern Theoretical Physics Conference, Chapel Hill, North Carolina, 1963.
[12] T. Ehrhardt, “A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities” in Recent Advances in Operator Theory (Groningen, 1998) , Oper. Theory Adv. Appl. 124 , Birkhäuser, Basel, 2001, 217-241. · Zbl 0993.47028
[13] M. E. Fisher and R. E. Hartwig, Toeplitz determinants: Some applications, theorems, and conjectures , Adv. Chem. Phys. 15 (1968), 333-353.
[14] A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Yu. Novokshenov, Painlevé Transcendents: The Riemann-Hilbert Approach , Math. Surveys Monogr. 128 , Amer. Math. Soc., Providence, 2006. · Zbl 1111.34001
[15] A. S. Fokas, A. R. Its, and A. V. Kitaev, The isomonodromy approach to matrix models in \(2\)D quantum gravity , Comm. Math. Phys. 147 (1992), 395-430. · Zbl 0760.35051
[16] A. S. Fokas, U. Muğan, and X. Zhou, On the solvability of Painlevé I, III and V , Inverse Problems 8 (1992), 757-785. · Zbl 0754.35107
[17] A. S. Fokas and X. Zhou, On the solvability of Painlevé II and IV , Comm. Math. Phys. 144 (1992), 601-622. · Zbl 0758.35058
[18] P. J. Forrester, N. E. Frankel, T. M. Garoni, and N. S. Witte, Finite one-dimensional impenetrable Bose systems: Occupation numbers , Phys. Rev. A (3) 67 , 043607, 2003.
[19] P. J. Forrester and N. S. Witte, Application of the \(\tau\)-function theory of Painlevé equations to random matrices: \(\mathrm P_{V}\), \(\mathrm P_{III}\), the LUE, JUE, and CUE , Comm. Pure Appl. Math. 55 (2002), 679-727. · Zbl 1029.34087
[20] A. Foulquié Moreno, A. Martinez-Finkelshtein, and V. L. Sousa, On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials , J. Approx. Theory 162 (2010), 807-831. · Zbl 1193.42099
[21] Y. V. Fyodorov, G. A. Hiary, and J. P. Keating, Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function , Phys. Rev. Lett. 108 (2012), art. ID 170601, 5 pp.
[22] Y. V. Fyodorov and J. P. Keating, Freezing transitions and extreme values: Random matrix theory, \(\zeta(1/2+it)\) and disordered landscapes , Philos. Trans. R. Soc. A 372 (2014), 20120503. · Zbl 1330.82028
[23] M. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension , J. Math. Phys. 1 (1960), 516-523. · Zbl 0098.21704
[24] A. Its and I. Krasovsky, “Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump” in Integrable Systems and Random Matrices , Contemp. Math. 458 , Amer. Math. Soc., Providence, 2008, 215-247. · Zbl 1163.15027
[25] M. Jimbo, T. Miwa, Y. Môri, and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent , Phys. D 1 (1980), 80-158. · Zbl 1194.82007
[26] J. P. Keating and N. C. Snaith, Random matrix theory and \(\zeta(1/2+it)\) , Comm. Math. Phys. 214 (2000), 57-89. · Zbl 1051.11048
[27] I. Krasovsky, Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant , Duke Math. J. 139 (2007), 581-619. · Zbl 1173.15012
[28] A. Lenard, Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons , J. Math. Phys. 5 (1964), 930-943.
[29] E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas, I: The general solution and the ground state , Phys. Rev. (2) 130 (1963), 1605-1616. · Zbl 0138.23001
[30] B. M. McCoy, The connection between statistical mechanics and quantum field theory , preprint, .
[31] B. M. McCoy, C. A. Tracy, and T. T. Wu, Painlevé functions of the third kind , J. Math. Phys. 18 (1977), 1058-1092. · Zbl 0353.33008
[32] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model , Harvard Univ. Press, Cambridge, Mass., 1973. · Zbl 1094.82500
[33] T. D. Schultz, Note on the one-dimensional gas of impenetrable point-particle bosons , J. Math. Phys. 4 (1963), 666-671.
[34] E. C. Titchmarsh, The Theory of Functions , 2nd ed., Oxford Univ. Press, Oxford, 1939. · Zbl 0022.14602
[35] C. A. Tracy, Asymptotics of a \(\tau\)- function arising in the two-dimensional Ising model , Comm. Math. Phys. 142 (1991), 297-311. · Zbl 0734.60106
[36] H. G. Vaidya and C. A. Tracy, One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature , J. Math. Phys. 20 (1979), 2291-2312.
[37] H. Widom, Toeplitz determinants with singular generating functions , Amer. J. Math. 95 (1973), 333-383. · Zbl 0275.45006
[38] T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region , Phys. Rev. B 13 (1976), 316-374.
[39] X. Zhou, The Riemann-Hilbert problem and inverse scattering , SIAM J. Math. Anal. 20 (1989), 966-986. · Zbl 0685.34021
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