## Duke Mathematical Journal

### Toeplitz determinants with merging singularities

#### Abstract

We study asymptotic behavior for the determinants of $n\times n$ Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance $2t\ge0$ from each other on the unit circle. We obtain large $n$ asymptotics which are uniform for $0\lt t\lt t_{0}$, where $t_{0}$ is fixed. They describe the transition as $t\to0$ between the asymptotic regimes of two singularities and one singularity. The asymptotics involve a particular solution to the Painlevé V equation. We obtain small and large argument expansions of this solution. As applications of our results, we prove a conjecture of Dyson on the largest occupation number in the ground state of a one-dimensional Bose gas, and a conjecture of Fyodorov and Keating on the second moment of powers of the characteristic polynomials of random matrices.

#### Article information

Source
Duke Math. J., Volume 164, Number 15 (2015), 2897-2987.

Dates
Revised: 23 October 2014
First available in Project Euclid: 1 December 2015

https://projecteuclid.org/euclid.dmj/1448980436

Digital Object Identifier
doi:10.1215/00127094-3164897

Mathematical Reviews number (MathSciNet)
MR3430454

Zentralblatt MATH identifier
1333.15018

#### Citation

Claeys, T.; Krasovsky, I. Toeplitz determinants with merging singularities. Duke Math. J. 164 (2015), no. 15, 2897--2987. doi:10.1215/00127094-3164897. https://projecteuclid.org/euclid.dmj/1448980436

#### References

• [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.
• [2] E. Basor, Asymptotic formulas for Toeplitz determinants, Trans. Amer. Math. Soc. 239 (1978), 33–65.
• [3] E. Basor, A localization theorem for Toeplitz determinants, Indiana Univ. Math. J. 28 (1979), 975–983.
• [4] A. Böttcher and B. Silbermann, Toeplitz operators and determinants generated by symbols with one Fisher–Hartwig singularity, Math. Nachr. 127 (1986), 95–123.
• [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.
• [6] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, Courant Lect. Notes Math. 3, Amer. Math. Soc., Providence, 1998.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [16] A. S. Fokas, U. Muğan, and X. Zhou, On the solvability of Painlevé I, III and V, Inverse Problems 8 (1992), 757–785.
• [17] A. S. Fokas and X. Zhou, On the solvability of Painlevé II and IV, Comm. Math. Phys. 144 (1992), 601–622.
• [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: $\rm P_{V}$, $\rm P_{III}$, the LUE, JUE, and CUE, Comm. Pure Appl. Math. 55 (2002), 679–727.
• [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.
• [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.
• [23] M. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension, J. Math. Phys. 1 (1960), 516–523.
• [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.
• [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.
• [26] J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta(1/2+it)$, Comm. Math. Phys. 214 (2000), 57–89.
• [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.
• [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.
• [30] B. M. McCoy, The connection between statistical mechanics and quantum field theory, preprint, arXiv:hep-th/9403084v2.
• [31] B. M. McCoy, C. A. Tracy, and T. T. Wu, Painlevé functions of the third kind, J. Math. Phys. 18 (1977), 1058–1092.
• [32] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model, Harvard Univ. Press, Cambridge, Mass., 1973.
• [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.
• [35] C. A. Tracy, Asymptotics of a $\tau$- function arising in the two-dimensional Ising model, Comm. Math. Phys. 142 (1991), 297–311.
• [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.
• [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.