## Duke Mathematical Journal

### Large deviations and the Lukic conjecture

#### Abstract

We use the large deviation approach to sum rules pioneered by Gamboa, Nagel, and Rouault to prove higher-order sum rules for orthogonal polynomials on the unit circle. In particular, we prove one half of a conjectured sum rule of Lukic in the case of two singular points, one simple and one double. This is important because it is known that the conjecture of Simon fails in exactly this case, so this article provides support for the idea that Lukic’s replacement for Simon’s conjecture might be true.

#### Article information

Source
Duke Math. J., Volume 167, Number 15 (2018), 2857-2902.

Dates
Revised: 16 May 2018
First available in Project Euclid: 3 October 2018

https://projecteuclid.org/euclid.dmj/1538532052

Digital Object Identifier
doi:10.1215/00127094-2018-0027

Mathematical Reviews number (MathSciNet)
MR3865654

Zentralblatt MATH identifier
06982209

#### Citation

Breuer, Jonathan; Simon, Barry; Zeitouni, Ofer. Large deviations and the Lukic conjecture. Duke Math. J. 167 (2018), no. 15, 2857--2902. doi:10.1215/00127094-2018-0027. https://projecteuclid.org/euclid.dmj/1538532052

#### References

• [1] G. Anderson, A. Guionnet, and O. Zeitouni, An Introduction to Random Matrices, Cambridge Stud. Adv. Math. 118, Cambridge Univ. Press, Cambridge, 2010.
• [2] G. Ben Arous and A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy, Probab. Theory Related Fields 108 (1997), 517–542.
• [3] J. Breuer, B. Simon, and O. Zeitouni, Large deviations and sum rules for spectral theory—A pedagogical approach, to appear in J. Spectr. Theory, preprint, arXiv:1608.01467v2 [math.PR].
• [4] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Appl. Math. 38, Springer, New York, 1998.
• [5] S. Denisov and S. Kupin, Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight, J. Approx. Theory 139 (2006), 8–28.
• [6] J.-D. Deuschel and D. W. Stroock, Large Deviations, Pure Appl. Math. 137, Academic Press, Boston, 1989.
• [7] E. Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ric. Mat. 7 (1958), 102–137.
• [8] F. Gamboa, J. Nagel, and A. Rouault, Sum rules via large deviations, J. Funct. Anal. 270 (2016), 509–559.
• [9] F. Gamboa, J. Nagel, and A. Rouault, Sum rules and large deviations for spectral measures on the unit circle, Random Matrices Theory Appl. 6 (2017), no. 1750005.
• [10] F. Gamboa, J. Nagel, and A. Rouault, Sum rules and large deviations for spectral matrix measures, preprint, arXiv:1601.08135v2 [math.PR].
• [11] L. Golinskii and A. Zlatoš, Coefficients of orthogonal polynomials on the unit circle and higher-order Szegő theorems, Constr. Approx. 26 (2007), 361–382.
• [12] D. J. Gross and E. Witten, Possible third-order phase transition in the large-$N$ lattice gauge theory, Phys. Rev. D 21 (1980), 446–453.
• [13] R. Killip and I. Nenciu, CMV: The unitary analogue of Jacobi matrices, Comm. Pure Appl. Math. 60 (2007), 1148–1188.
• [14] R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. (2) 158 (2003), 253–321.
• [15] S. Kupin, On a spectral property of Jacobi matrices, Proc. Amer. Math. Soc. 132 (2004), 1377–1383.
• [16] A. Laptev, S. Naboko, and O. Safronov, On new relations between spectral properties of Jacobi matrices and their coefficients, Comm. Math. Phys. 241 (2003), 91–110.
• [17] M. Lukic, On a conjecture for higher-order Szegő theorems, Constr. Approx. 38 (2013), 161–169.
• [18] M. Lukic, On higher-order Szegő theorems with a single critical point of arbitrary order, Const. Approx. 44 (2016), 283–296.
• [19] F. Nazarov, F. Peherstorfer, A. Volberg, and P. Yuditskii, On generalized sum rules for Jacobi matrices, Int. Math. Res. Not. IMRN 2005, no. 3, 155–186.
• [20] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa (3) 13 (1959), 115–162.
• [21] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Amer. Math. Soc. Colloq. Publ. 54, Amer. Math. Soc., Providence, 2005.
• [22] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, Amer. Math. Soc. Colloq. Publ. 54, Amer. Math. Soc., Providence, 2005.
• [23] B. Simon, Szegő’s Theorem and Its Descendants: Spectral Theory for $L^{2}$ Perturbations of Orthogonal Polynomials, Princeton Univ. Press, Princeton, 2011.
• [24] B. Simon, Harmonic Analysis: A Comprehensive Course in Analysis, Part 3, Amer. Math. Soc., Providence, 2015.
• [25] B. Simon, Real Analysis: A Comprehensive Course in Analysis, Part 1, Amer. Math. Soc., Providence, 2015.
• [26] B. Simon and A. Zlatoš, Higher-order Szegő theorems with two singular points, J. Approx. Theory 134 (2005), 114–129.
• [27] M. E. Taylor, Partial Differential Equations, III: Nonlinear Equations, 2nd ed., Appl. Math. Sci. 117, Springer, New York, 2011.
• [28] S. Verblunsky, On positive harmonic functions, Proc. London Math. Soc. (2) 40 (1935), 290–320.
• [29] J. Yan, An algebra model for the higher-order sum rules, to appear in J. Constr. Approx. preprint, arXiv:1706.07925v3 [math.SP].