Class Note for PHYS 142 at UA 2
Class Note for PHYS 142 at UA 2
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Date Created: 02/06/15
LECTURE 19 THE CONNECTION BETWEEN ORTHOGONAL POLYNOMIALS AND RIEMANNiHILBERT PROBLEMS Lecture plan We will discuss the connection between orthogonal polynomials and RiemanniHilbert prob lems the aim being to compute the behavior of KNzy for N A 00 for a wide family of ensembles of random matrices SUMMARY OF THE PAST 3 LECTURES Momar has explained the following important results 0 We can consider random symmetric matrices rather than random Hermitian matrices and there is still a determinantal formula for eigenvalue statistics at least for the case of the Gaussian Or thogonal Ensemble This can be generalized to nonGaussian measures as well and can also be generalized to random symplectic matrices o The orthogonal polynomial kernel N71 lt0 Imamarmrm zmmmm 0 satis es the amazing ChristoffelDarboux formula 2 KN17y 6V Vy ltPNIPN719 PNyPN71Igt 7 H N z 7 y where for each natural number j Hj is the leading coef cient of the jth orthogonal polynomial 1011 H jf39w 0 There is a limit law for the largest eigenvalue This limit law is obtained by computing asymptotics for the orthogonal polynomial kernel and you can derive that in the appropriate limit the kernel converges to a new kernel built out of the Airy function Recall that to compute statistical properties of eigenvalues we have seen that the orthogonal polynomial kernel KNzy is the fundamental quantity For example using a great deal of trickery we were able to show that the Fredholm determinant 3 Ha b t det 1 7 mmwm is a generating function for certain number statistics relative to the interval a b the most fundamental being 4 Prob no eigs in a b Hab l and one application of this is the following 5 Prob Amax lt a HaOol Our interest is in studying the behavior of this function as N A 00 and the above formulae admit such analysis if the operator KN converges in the Trace norm then the function Habt also converges ORTHOGONAL POLYNOMIALS AND RIEMANNiHILBERT PROBLEMS The following RiemannHilbert problem 1 is known to characterize the polynomials pltNgt orthogonal with J respect to e NW RiemannHilbert Problem 1 Find a 2 X 2 matrix Az Az nN with the properties Analyticity Az is analytic for 2 E C R and takes continuous boundary values A I A as 2 tends to z with z E R andz 6 3 2 E 3 Jump Condition The boundary values are connected by the relation 1 eina mmLm 0 l Normalization The matrix Az is normalized at 2 00 as follows 2 0 7 21320 Az lit 0 2 It was discovered in 1 that RiemannHilbert Problem 1 characterizes polynomials orthogonal with respect to dlz I e NW dzi The connection between these orthogonal polynomials and the solution of Riemann Hilbert Problem 1 is the following 1 l pnse NV5 7 Z 7 ds H533 ailH533 R H 8 A z 7NV5 Zm ig 71711071719 Hgiii nil d5 R S i Z This relationship provides a useful avenue for asymptotic analysis of the orthogonal polynomials in the limit n A 00 it is suf cient to carry out a rigorous asymptotic analysis of RiemannHilbert Problem 1 In class we will present a proof that 8 represents the unique solution to RiemanniHilbert ll REPRESENTING THE KERNEL The kernel KN I y can be represented directly in terms of the solution of the RiemanniHilbert problem 1 VmVyY11I321y Edam119 9 KNI79 17y elltvltxgtvltygtgt 1 71 1 e 2 2m17y01YyYzlt0 i REFERENCES 1 AV Fokas AV lts and AV VV Kitaev Discrete Painleve equations and their appearance in quantum gravity Commun Math Phi5 142 3137344 1991
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