Abstract
This paper focuses on the characteristics of the Hankel determinant generated by a modified singularly Gaussian weight. The weight function is defined as:
where \(\alpha >1\) and \(t\in (0,1)\) are parameters. Using ladder operator techniques, we derive a series of difference formulas for the auxiliary quantities and recurrence coefficients. We present the difference equations for the recurrence coefficients and the logarithmic derivative of the Hankel determinant. We then use the “t-dependence" to obtain the differential identities satisfied by the auxiliary quantities and the logarithmic derivative of the Hankel determinant. To obtain the large n asymptotic expressions of the recurrence coefficients, we use the Coulomb fluid method together with the related difference equations, which depend on n either being odd or even. We also obtain the reduction forms of the second-order differential equations satisfied by the orthogonal polynomials generated by this weight. Two special cases coincide with the bi-confluent Heun equation and the double confluent Heun equation, respectively. Finally, we calculate the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by this weight. Our result not only covers the classical result of Szegö (Trans Am Math Soc 40:450–461, 1936) but also determines our next research direction.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
References
Basor, E.L., Chen, Y., Ehrhardt, T.: Painlevé V and time-dependent Jacobi polynomials. J. Phys. A: Math. Theor. 43, 015204 (2010)
Bonan, S., Clark, D.S.: Estimates of the orthogonal polynomials with weight \(exp(-x^m)\), \(m\) an even positive integer. J. Approx. Theory 46, 408–410 (1986)
Bonan, S., Nevai, P.: Orthogonal polynomials and their derivatives I. J. Approx. Theory 40, 134–147 (1984)
Chen, M., Chen, Y.: Singular linear statistics of the Laguerre unitary ensemble and Painlevé III: Double scaling analysis. J. Math. Phys. 56, 063506 (2015)
Chen, Y., Dai, D.: Painlevé V and a Pollaczek–Jacobi type orthogonal polynomials. J. Approx. Theory 162, 2149–2167 (2010)
Chen, Y., Ismail, M.E.H.: Thermodynamic relations of the Hermitian matrix ensembles. J. Phys. A: Math. Gen. 30, 6633–6654 (1997)
Chen, Y., Ismail, M.E.H.: Jacobi polynomials from compatibility conditions. Proc. Am. Math. Soc. 133, 465–472 (2005)
Chen, Y., Its, A.: Painlevé III and a singular linear statistics in Hermitian random matrix ensembles I. J. Approx. Theory 162, 270–297 (2010)
Chen, Y., Lawrence, N.: On the linear statistics of Hermitian random matrices. J. Phys. A: Math. Gen. 31, 1141–1152 (1998)
Chen, Y., Lawrence, N.: Small eigenvalues of large Hankel matrices. J. Phys. A: Math. Gen. 32, 7305–7315 (1999)
Chen, Y., Lubinsky, D.S.: Smallest eigenvalues of Hankel matrices for exponential weights. J. Math. Anal. Appl. 293, 476–495 (2004)
Chen, Y., Mckay, M.R.: Coulomb fluid, Painlevé transcendents and the information theory of MIMO systems. IEEE Trans. Inform. Theory 58, 4594–4634 (2012)
Chen, Y., Filipuk, G., Zhan, L.: Orthogonal polynomials, asymptotics and Heun equation. J. Math. Phys. 60, 113501 (2019)
Clarkson, P.A., Jordaan, K.: The relationship between semiclassical Laguerre polynomials and the fourth Painlevé equation. Constr. Approx. 39, 223–254 (2014)
Clarkson, P.A., Jordaan, K., Kelil, A.: A generalized Freud weight. Stud. Appl. Math. 136, 288–320 (2016)
Dai, D., Zhang, L.: Painlevé VI and Hankel determinants for the generalized Jacobi weight. J. Phys. A: Math. Theor. 43, 055207 (2010)
Dyson, F.J.: Statistical theory of the energy levels of complex systems I. J. Math. Phys. 3, 140–175 (1962)
Fokas, A., Its, A., Kitaev, A.: Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys. 142, 313–344 (1991)
Forrester, P.J., Witte, N.S.: Application of the \(\tau -\)function theory of Painlevé equations to random matrices: P\(_{V}\), P\(_{III}\), the LUE, JUE and CUE. Commun. Pure Appl. Math. 55, 679–727 (2002)
Freud, G.: On the coefficients in the recursion formulae of orthogonal polynomials, In: Proceeding of the Royal Irish Academy. Section A: Mathematical and Physical Sciences; Royal Irish Academy: Dublin, Ireland, pp. 1–6 (1976)
Gradshteyn, I. S., Ryzhik, I. M.: Table of Integrals, Series, and Products, 7th ed. (Elsevier; Academic Press, Amsterdam, 2007), p. xlviii+1171, translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX)
Han, P., Chen, Y.: The recurrence coefficients of a semi-classical Laguerre polynomials and the large \(n\) asymptotics of the associated Hankel determinant. Randon Matrices 6, 1740002 (2017)
Kasuga, T., Sakai, R.: Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121, 13–53 (2003)
Kelil, A., Appadu, A.: On semi-classical orthogonal polynomials associated with a modified sextic Freud-type weight. Mathematics 8, 1250 (2020)
Lyu, S., Griffin, J., Chen, Y.: The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble. J. Nonlinear Math. Phys. 26(1), 24–53 (2019)
Min, C., Lyu, S., Chen, Y.: Painlevé III\(^{\prime }\) and the Hankel determinant generated by a singularly perturbed Gaussian weight. Nucl. Phys. B 936, 169–188 (2018)
Ohyama, Y., Kawamuko, H., Sakai, H., Okamoto, K.: Studies on the Painlevé equations, V, third Painlevé equations of special type \(P_{III}\) (\(D_7\)) and \(P_{III}\) (\(D_8\)). J. Math. Sci. Univ. Tokyo 13, 145–204 (2006)
Pólya, G., Szegö, G.: Problems and Theorems in Analysis I. Spring, Berlin (1978)
Riesz, F., Nagy, B.S.: Functional Analysis. Blackie &Son Limited, London (1956)
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften 316. Springer, Berlin (1997)
Shohat, J.: A differential equation for orthogonal polynomials. Duke Math. J. 5, 401–417 (1939)
Szegö, G.: On some Hermitian forms associated with two given curves of the complex plane. Trans. Am. Math. Soc. 40, 450–461 (1936)
Wang, D., Zhu, M., Chen, Y.: On semiclassical orthogonal polynomials associated with a Freud-type weight. Math. Methods Appl. Sci. 43, 5295–5313 (2020)
Wang, D., Zhu, M., Chen, Y.: The smallest eigenvalue of large Hankel matrices associated with a singularly perturbed Gaussian weight. Proc. Am. Math. Soc. 150, 153–160 (2022)
Xu, S.X., Dai, D., Zhao, Y.Q.: Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight. J. Approx. Theory 192, 1–18 (2015)
Yu, J., Li, C., Zhu, M., Chen, Y.: Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalues. J. Math. Phys. 63, 063504 (2022)
Zhu, M., Chen, Y.: On properties of a deformed Freud weight. Random Matrices 8, 1950004 (2019)
Zhu, M., Chen, Y., Emmart, N., Weems, C.: The smallest eigenvalue of large Hankel matrices. Appl. Math. Comput. 334, 375–387 (2018)
Zhu, M., Chen, Y., Li, C.: The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight. J. Math. Phys. 61, 073502 (2020)
Acknowledgements
D.Wang would like to acknowledge the support of Changzhou University for Grant No. ZMF22020116. M. Zhu was supported by the National Natural Science Foundation of China under Grant No. 12201333, the Natural Science Foundation of Shandong Province under Grant No. ZR2021QA034, and the Youth Entrepreneurship and Innovation Technology Support Program of Shandong Provincial Higher Education Institutions under Grant No. 2023KJ135.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with this work.
Additional information
Communicated by F W Nijhoff.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Wang, D., Zhu, M. Discrete, Continuous and Asymptotic for a Modified Singularly Gaussian Unitary Ensemble and the Smallest Eigenvalue of Its Large Hankel Matrices. Math Phys Anal Geom 27, 5 (2024). https://doi.org/10.1007/s11040-024-09477-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-024-09477-w