Class Note for PHYS 142 at UA
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Date Created: 02/06/15
LECTURE 15 EIGENVALUE STATISTICS AND THE CONNECTION BETWEEN ORTHOGONAL POLYNOMIALS AND RIEMANNiHILBERT PROBLEMS Lecture plan We will discuss more detailed statistical information about the number of eigenvalues in an interval Then we will discuss the connection between orthogonal polynomials and RiemanniHilbert problems the aim being to compute the behavior of KNzy for N A 00 SUMMARY OF THE PREVIOUS LECTURE Recall that to compute the probability that there are no eigenvalues in an interval ab we considered the characteristic function of the interval lt1 xltzgt mm 5 if my Then we found the following formula N 2 Prob no eigs in ab E H 17 xj F1 which eventually led to the function N 3 HabtlE Human 7 F1 and using a great deal of trickery we were able to show that this has a representation in terms of a Fredholm determinant 4 Ha b t det17tKNi Expanding the product we nd N 5 Ha7b7t1itE ZXW HZE Z Xi1xi2 11 1 i1lti2 N N 1 87 2 H Wu j3 15i1ltltigNk1 Our interest is in studying the behavior of this function as N A 00 and the above formula lends itself to such analysis if the operator KN converges in the Trace norm then the function Ha b t also convergesl LARGE ST EIGENVALUE A special version of the probability that there are no eigenvalues in an interval is the case a b a 00 for in this case we have 6 Prob Amax lt a Haoo1 NUMBER PROBABILITIES It is also possible using the same considerations to study the probability that there are n eigenvalues in the interval a b N 7 Prob eigs 6 ab n E E H X M2 H 1 X AllaD 1 i1ltltin N 1 g 1 if if 221 7lt1gtndnHlt w 7 0397 7 1 11 On the other hand7 we can also ask for higher statistics concerning the number of eigenvalues within the interval a7 b Let us de ne the random variable a7 b7 M to be the number of eigenvalues in a7 b 8 aybyMeigS 601717 We have already observed that 9 1Eab 7Habt t0 Let us compute the variance of this random variable lt10gtvar lt a b M E lt a M 7 E lt a b Mm E lt MM 7 1 ab M Now we can compute the rst expectation by our usual trickery N E aiMw RN axonxom det KNltAmAngtgtNdeNA N l l ZxontdetKNltAmAngtgtNXNdNA2 Z xltAjgtxltAkgtidetKNltAmAngtgtNdeNA N N W N 11 1 JltkSN b b b KNAAdA detKNAmAn2X2dA1dA2 Combining this with 107 we nd lt11gtvar lt WM b b b b b 2 Kva7 Ad 5 det KNOWL An2x2 dAldAg 7 KN1 A1KN2 A2d1d2 So apparently the nal claim from Lecture 14 was wrong But here is a corrected formula 12 VaraybyM d2 7 Hab7 tH ab7 t 7H abt2 y 7 logHabt 10 Ha7b7t2 H ab0 7 Ha7b702 t0 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 eiNVw 13gt Altzgt A z lt gt a 0 1 Normalization The matrix Az is normalized at 2 00 as follows 2 0 l4 lim Az ll zaoo 0 Zn 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 p 1 pnse NV5dS N n 27wth 1R 5 2 nn 15gt Altzgt d3 N Pn71se NV5 N 72W2Hn71 z71p i1lt21gt i nilmil 37 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 15 represents the unique solution to RiemanniHilbert 1 REFERENCES 1 AV Fokas AV Its and AV VV Kitaev Discrete Pajnleve equations and their appearance in quantum gravity Commun Math Phys 142 3137344 1991
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