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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. (English) Zbl 0944.42013
In this important and interesting paper, the authors again illustrate the awesome power of the Deift-Zhou steepest descent method associated with Riemann-Hilbert problems. Let $$V: \mathbb{R}\to\mathbb{R}$$ be real-valued and analytic on $$\mathbb{R}$$, with $\lim_{|x|\to\infty} V(x)/\log(1+ x^2)=\infty.$ For $$n\geq 1$$, let $$\{p_k(x; n)\}^\infty_{k= 0}$$ denote the sequence of orthonormal polynomials for the weight $$\exp(-nV)$$, so that $\int p_k(x; n)p_j(x; n)\exp(- nV(x)) dx= \delta_{jk}\quad\forall j,k\geq 0.$ The authors derive Plancherel-Rotach asymptotics for $$p_n(z; n)$$ as $$n\to\infty$$, valid in every region of the plane. The precision of the asymptotics on and off the real line is remarkable. The proofs involve the Fokas-Its-Kitaev identify for the orthonormal polynomials as solutions of a Riemann-Hilbert problem, followed by the Deift-Zhou steepest descent technique. The details are intricate, but are clearly presented, and the main ideas are summarized to guide the reader through the proofs.
As an application, the authors prove universality limits that arise in random matrix theory. These involve the weighted reproducing kernel $K_n(x,y)= e^{-{n\over 2}(V(x)+ V(y))} \sum^{n- 1}_{j= 0} p_j(x; n)p_j(y; n)$ and have the form ${1\over n\psi(a)} K_n\Biggl(a+ {s\over n\psi(a)}, a+{t\over n\psi(a)}\Biggr)= {\sin\pi(s- t)\over \pi(s- t)}+ O\Biggl({1\over n}\Biggr),$ uniformly for $$s$$, $$t$$ in compact subsets of $$\mathbb{R}$$. Here $$a$$ lies in a subinterval of the support of the equilibrium measure $$\mu_V$$ associated with the field $$V$$, and $$\psi$$ is the density of that equilibrium measure.

##### MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 15B52 Random matrices (algebraic aspects) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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##### References:
 [1] ; Handbook of mathematical functions, with formulas, graphs, and mathematical tables. Dover, New York, 1966. [2] Beals, Comm Pure Appl Math 37 pp 39– (1984) · Zbl 0514.34021 · doi:10.1002/cpa.3160370105 [3] ; ; Direct and inverse scattering on the line. Mathematical Surveys and Monographs, 28. American Mathematical Society, Providence, R.I., 1988. · doi:10.1090/surv/028 [4] ; Asymptotics of orthogonal polynomials and universality in matrix models. Preprint, 1996. [5] ; Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, 3. Birkhäuser, Basel-Boston, 1981. · doi:10.1007/978-3-0348-5492-4 [6] Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. Courant Institute, New York, 1999. [7] Deift, Ann of Math (2) 146 pp 149– (1997) · Zbl 0936.47028 · doi:10.2307/2951834 [8] ; ; New results for the asymptotics of orthogonal polynomials and related problems via the Lax-Levermore method. Recent advances in partial differential equations, Venice 1996, 87-104. Proc Sympos Appl Math, 54. Amer Math Soc, Providence, R.I., 1998. · doi:10.1090/psapm/054/1492693 [9] Deift, Internat Math Res Notices 16 pp 759– (1997) · Zbl 0897.42015 · doi:10.1155/S1073792897000500 [10] Deift, Comm Pure Appl Math 52 (1999) · Zbl 1026.42024 · doi:10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.0.CO;2-# [11] Deift, Internat Math Res Notices 6 pp 286– (1997) · Zbl 0873.65111 · doi:10.1155/S1073792897000214 [12] Deift, Ann of Math (2) 137 pp 295– (1993) · Zbl 0771.35042 · doi:10.2307/2946540 [13] Deift, Comm Pure Appl Math 48 pp 277– (1995) · Zbl 0869.34047 · doi:10.1002/cpa.3160480304 [14] ; Long time behavior of the non-focusing nonlinear pasturSchrödinger equation?a case study. NS: Lectures in Mathematical Sciences, 5. University of Tokyo, 1994. [15] Dubrovin, Uspekhi Mat Nauk 36 pp 11– (1981) [16] Theta-functions and nonlinear equations. (Russian) Russ Math Surv 36:2 (1981), 11-92. [17] Ercolani, Comm Math Phys 183 pp 119– (1997) · Zbl 0865.35115 · doi:10.1007/BF02509798 [18] ; ; The KdV zero dispersion limit via Dirichlet spectra and Weyl functions. Preprint, 1997. [19] ; Riemann surfaces. Graduate Texts in Mathematics, 71. Springer-Verlag, New York?Berlin, 1992. · Zbl 0764.30001 · doi:10.1007/978-1-4612-2034-3 [20] Fokas, Comm Math Phys 142 pp 313– (1991) · Zbl 0742.35047 · doi:10.1007/BF02102066 [21] Fokas, Comm Math Phys 147 pp 395– (1992) · Zbl 0760.35051 · doi:10.1007/BF02096594 [22] Gaudin, Nucl Phys 25 pp 447– (1961) · Zbl 0107.44605 · doi:10.1016/0029-5582(61)90176-6 [23] ; The isomonodromic deformation method in the theory of Painlevé equations. Lecture Notes in Mathematics, 1191. Springer-Verlag, Berlin?New York, 1986. · Zbl 0592.34001 · doi:10.1007/BFb0076661 [24] Johansson, Duke Math J 91 pp 151– (1998) · Zbl 1039.82504 · doi:10.1215/S0012-7094-98-09108-6 [25] ; Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, R.I., 1999. · Zbl 0958.11004 [26] Lax, Comm Pure Appl Math 36 pp 253– (1983) · Zbl 0532.35067 · doi:10.1002/cpa.3160360302 [27] Lax, Comm Pure Appl Math 36 pp 571– (1983) · Zbl 0527.35073 · doi:10.1002/cpa.3160360503 [28] Lax, Comm Pure Appl Math 36 pp 809– (1983) · Zbl 0527.35074 · doi:10.1002/cpa.3160360606 [29] Strong asymptotics for extremal errors and polynomials associated with Erdös-type weights. Pitman Research Notes in Mathematics Series, 202. Longman, Harlow; copublished in the United States with John Wiley, New York, 1989. [30] Random matrices. Second edition. Academic Press, Boston, 1991. [31] Mehta, Nuclear Phys 18 pp 420– (1960) · Zbl 0107.35702 · doi:10.1016/0029-5582(60)90414-4 [32] Nevai, SIAM J Math Anal 15 pp 1177– (1984) · Zbl 0566.42016 · doi:10.1137/0515092 [33] Spectral and probabilistic aspects of matrix models. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 207-242 Math Phys Stud, 19. Kluwer, Dordrecht, 1996. · doi:10.1007/978-94-017-0693-3_10 [34] Pastur, J Statist Phys 86 pp 109– (1997) · Zbl 0916.15009 · doi:10.1007/BF02180200 [35] Strong asymptotics for orthogonal polynomials. Methods of approximation theory in complex analysis and mathematical physics (Leningrad, 1991), 71-97. Lecture Notes in Math, 1550, Springer, Berlin, 1993. · doi:10.1007/BFb0117475 [36] ; Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich], New York?London, 1978. · Zbl 0401.47001 [37] One-dimensional perturbations of a differential operator, and the inverse scattering problem. (Russian) Problems in mechanics and mathematical physics (Russian), 279-296, 298. Izdat. ?Nauka,? Moscow, 1976. [38] ; Logarithmic potentials with external fields. Appendix B by Thomas Bloom. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 316. Springer, Berlin, 1997. · Zbl 0881.31001 · doi:10.1007/978-3-662-03329-6 [39] Sheen, J Approx Theory 50 pp 232– (1987) · Zbl 0617.42017 · doi:10.1016/0021-9045(87)90021-9 [40] Trace ideals and their applications. London Mathematical Society Lecture Note Series, 35. Cambridge University Press, Cambridge?New York, 1979. · Zbl 0423.47001 [41] Orthogonal Polynomials. American Mathematical Society Colloquium Publications, 23. American Mathematical Society, Providence, R.I., 1939. · doi:10.1090/coll/023 [42] Zhou, SIAM J Math Anal 20 pp 966– (1989) · Zbl 0685.34021 · doi:10.1137/0520065 [43] Riemann-Hilbert problems and integrable systems. Lecture notes, preprint.
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