Abstract
In this short article, we propose a full large N asymptotic expansion of the probability that the \(m{\text {th}}\) power of a random unitary matrix of size N has all its eigenvalues in a given arc-interval centered in 1 when N is large. This corresponds to the asymptotic expansion of a Toeplitz determinant whose symbol is the indicator function of several intervals having a discrete rotational symmetry. This solves and improves a conjecture left opened by the author in Marchal (Lett Math Phys 110:211–258, 2020). It also provides a rare example of the explicit computation of a full asymptotic expansion of a genus \(g>0\) classical spectral curve, including the oscillating non-perturbative terms, using the topological recursion.
Similar content being viewed by others
Notes
Note that the definition of the Toeplitz determinant \(D_N(m,\epsilon )\) remains unchanged by any global rotation of the angles (i.e., \(\theta _i\mapsto \theta _i+\alpha \) for all \(i\in \llbracket 1,N\rrbracket \)). In particular, in [37], for m even, all angles where shifted by \(\frac{2\pi }{2m}\) so that the function \(\theta \mapsto \tan \frac{\theta }{2}\) could be applied on all intervals defining \(I_m(\epsilon )\).
Note that \(n_1\) and \(n_2\) depends on N (\(n_2\) a periodic function of N with period m while \(n_1=\left\lfloor \frac{N}{m}\right\rfloor \)) but we shall not write down the dependence explicitly for clarity.
References
Basor, E.L., Morrison, K.E.: The Fisher–Hartwig conjecture and Toeplitz eigenvalues. Linear Algebra Appl. 202, 129–142 (1994)
Blackstone, E., Charlier, C., Lenells, J.: Oscillatory asymptotics for the Airy kernel determinant on two intervals. Int. Math. Res. 2022(4), 2636–2687 (2020)
Blackstone, E., Charlier, C., Lenells, J.: The Bessel kernel determinant on large intervals and Birkhoff’s ergodic theorem. arXiv:2101.09216 (2021)
Blackstone, E., Charlier, C., Lenells, J.: Gap probabilities in the bulk of the Airy process. Random Matrices Theory Appl. 11, 1–30 (2021)
Borodin, A., Okounkov, A.: A Fredholm determinant formula for Toeplitz determinants. Integr. Equ. Oper. Theory. 37, 386–396 (2000)
Borot, G., Guionnet, A.: Asymptotic expansion of \(\beta \) matrix models in the multi-cut regime. arXiv:1303.1045 (2013)
Borot, G., Guionnet, A.: Asymptotic expansion of \(\beta \) matrix models in the one-cut regime. Commun. Math. Phys. 317, 447–483 (2013)
Borot, G., Guionnet, A., Kozlowski, K.: Large \(n\) asymptotic expansion for mean field models with Coulomb gas interaction. Int. Math. Res. 2015, 10451–10524 (2015)
Charlier, C.: Large gap asymptotics on annuli in the random normal matrix model. Mathematische Annalen Math. Ann. (2021)
Charlier, C., Claeys, T.: Asymptotics for Toeplitz determinants: perturbation of symbols with a gap. J. Math. Phys. 56, 022705 (2015)
Charlier, C., Claeys, T.: Thinning and conditioning of the Circular Unitary Ensemble. Random Matrices Theory Appl. 6, 1750007 (2017)
Charlier, C., Fahs, B., Webb, C., Wong, M.D.: Asymptotics of Hankel determinants with a multi-cut regular potential and Fisher-Hartwig singularities. arXiv:2111.08395 (2021)
Chekhov, L.O., Eynard, B., Marchal, O.: Topological expansion of \(\beta \)-ensemble model and quantum algebraic geometry in the sectorwise approach. Theor. Math. Phys. 166, 141–185 (2011)
Costin, O., Dunne, G.: Convergence from divergence. J. Phys. Math. Theor. 51, 1–10 (2018)
Costin, O., Dunne, G.: Resurgent extrapolation: rebuilding a function from asymptotic data. Painlev é i. J. Phys. Math. Theor. 52, 445205 (2019)
Deift, P., Its, A., Krasovsky, I.: Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher–Hartwig singularities. Ann. Math. 174, 1243–1299 (2011)
Deift, P., Its, A., Krasovsky, I.: On the asymptotics of a Toeplitz determinant with singularities. Random Matrices 65, 93–146 (2014)
Deift, P., Its, A., Krasovsky, I., Zhou, X.: The Widom–Dyson constant for the gap probability in random matrix theory. J. Comput. Appl. Math. 202, 26–47 (2007)
Deift, P., Its, A., Zhou, X.: A Riemann–Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. 146, 149–235 (1997)
Diaconis, P., Shahshahani, M.: On the Eigenvalues of Random Matrices. J. Appl. Probab. 31, 49–62 (1994)
Duits, M., Johansson, K.: Powers of large random unitary matrices and Toeplitz determinants. Trans. Am. Math. Soc. 3, 1169–1180 (2010)
Duits, M., Kozhan, R.: Relative Szegö asymptotics for Toeplitz determinants. Int. Math. Res., 266, 5441–5496 (2017)
Dunin-Barkowski, P., Norbury, P., Orantin, N., Popolitov, A., Shadrin, S.: Dubrovin’s superpotential as a global spectral curve. J. Inst. Math. Jussieu 18, 449–497 (2019)
Eynard, B., Garcia-Failde, E., Marchal, O., Orantin, N.: Quantization of classical spectral curves via topological recursion. arXiv:2106.04339, (2021)
Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1, 347–452 (2007)
Fahs, B.: Double scaling limits of Toeplitz, Hankel and Fredholm determinants. PhD thesis, Université Catholique de Louvain (2017)
Fahs, B., Krasovsky, I.: Sine-kernel determinant on two large intervals. arXiv:2003.08136 (2020)
Fisher, M.E., Hartwig, R.E.: Toeplitz determinants, some applications, theorems and conjectures. Adv. Chem. Phys. 15, 333–353 (1968)
Golinskii, B., Ibragimov, I.: On Szegö’s limit theorem. Izv. Akad. Nauk SSSR Ser. Mat. 35, 408–427 (1971)
Ibragimov, I.: On a theorem of Szegö. Mat. Zametki 3, 693–702 (1968)
Johansson, K.: On Szegö asymptotic formula for Toeplitz determinants and generalizations. Bull. Sci. Math. 112, 257–304 (1988)
Krasovsky, I.: Asymptotics for Toeplitz determinants on a circular arc. arXiv:0401256 (2006)
Krasovsky, I.: Aspects of Toeplitz determinants. Prog. Probab. 64, 305–324 (2011)
Krasovsky, I., Maroudas, T.H.: Airy-kernel determinant on two large intervals. arXiv:2108.04495 (2021)
Marchal, O.: One-cut solution of the \(\beta \)-ensembles in the Zhukovsky variable. J. Stat. Mech. Theory Exp. 2011, P01011 (2011)
Marchal, O.: Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices. Random Matrices Theory Appl. 4, 1–60 (2014)
Marchal, O.: Asymptotic expansions of some Toeplitz determinants via the topological recursion. Lett. Math. Phys. 110, 211–258 (2020)
Marchal, O., Orantin, N.: Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the sl2 case. J. Math. Phys. 61, 061506 (2020)
Marchal, O., Orantin, N.: Quantization of hyper-elliptic curves from isomonodromic systems and topological recursion. J. Geom. Phys. 71, 104407 (2021)
Mehta, M.L.: Random Matrices, volume 142 of Pure and Applied Mathematics, 3rd edn. Elsevier Academic Press, Amsterdam (2004)
Nikolaev, N.: Exact solutions for the singularly perturbed Riccati equation and exact WKB analysis. Nagoya Math. J. 250, 434–469 (2023). https://doi.org/10.1017/nmj.2022.38
Nikolaev, N.: Existence and uniqueness of exact WKB solutions for second-order singularly perturbed linear ODEs. Commun. Math. Phys. 400(1), 463–517 (2023)
Nikolaev, N.: Exact Perturbative Existence and Uniqueness Theorem. arXiv:2201.04526 (2022)
Szegö, G.: Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion. Math. Ann. 76, 490–503 (1915)
Szegö, G.: On certain Hermitian forms associated with the Fourier series of a positive function. Communications et Séminaires Mathématiques de l’Université de Lund, pp. 228–238 (1952)
Toeplitz, O.: Zur Transformation der Scharen bilinearer Formen von unendlichvielen Veränderlichen. Nachr. Ges. Wiss. Göttingen 1907, 110–115 (1907)
Toeplitz, O.: Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veränderlichen. Math. Ann. 70, 351–376 (1911)
Widom, H.: Strong Szegö limit theorem on circular arcs. Indiana Univ. Math. J. 21, 277–283 (1971)
Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants. Adv. Math. (N Y) 13, 284–322 (1974)
Acknowledgements
I would like to thank C. Charlier for suggesting the reference to B. Fahs PhD. thesis and several other references. I would like to thank B. Fahs for fruitful discussions and N. Orantin for suggesting reference [23].
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A: Topological recursion details in the one-cut case
Appendix A: Topological recursion details in the one-cut case
In this section, we list the intermediate steps obtained in the computation of the topological recursion of [25] applied to the classical spectral curve of genus 0 given by (2.6) that may be parametrized globally as:
with the global involution \(\bar{z}=\frac{1}{z}\). The parametrization implies that there are two simple branchpoints at \(z=1\) and \(z=-1\). The definition of the topological recursion in this particular case is presented in Appendix B of [37]. It provides the following intermediate steps:
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Marchal, O. Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices. Lett Math Phys 113, 78 (2023). https://doi.org/10.1007/s11005-023-01700-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11005-023-01700-z