Class Note for PHYS 142 at UA
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Date Created: 02/06/15
LECTURE 23 RIEMANNiHILBERT PROBLEMS AND SINGULAR INTEGRAL EQUATIONS AND OUTLINE OF ASYMPTOTIC CALCULATION Lecture plan The goal of this lecture is to explain the steps involved with computing the asymptotic anal ysis of the RiemanniHilbert problem associated to orthogonal polynomials Having de ned the equilibrium measure we will introduce the rst transformation of the RiemanniHilbert problem and then proceed with an outline of the asymptotic analysis USING EQUILIBRIUM MEASURES Recall that the equilibrium measure is de ned as the unique minimizer in 1 M1R Probability measures on R of the functional 2 g M1 A 70000 M gt gt R2 loglz 7 ylil duzduy R Hg r dMII In the previous lecture we discussed the various origins of this variational problem and how it relates to orthogonal polynomials and random matrix theory In this lecture we will require later the following properties of the equilibrium measure 0 The equilibrium measure Itquot is ac wrt Lebesgue measure ffzdu 31 fzw zdz and 1 7 371 3 gM Wdu z 6 711 0 There is a constant Z so that for all z 6 711 the equilibrium measure satis es lt4 2 lloglz 7 yldWy 7 was z o For I E R 711 the following holds lt5 2 lloglz 7 yldWy 7 was lt z FIRST TRANSFORMATION OF THE RIEMANNiHILBERT PROBLEM Recall from the previous lecture that we have following RiemannHilbert problem which is known to characterize the polynomials pgN orthogonal with respect to e NW I I 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 e7NVd lt6 Altzgt A7 I 0 I Normalization The matrix Az is normalized at 2 00 as follows 2 0 7 lim Az ll zaoo The connection between these orthogonal polynomials and the solution of RiemannHilbert Problem 1 is the following 1 1 pnse NV5 7107i 2 ds i533 2mm R s 7 z 8 152 7NV 5 27ri illi1niipnil2 Hlijiimil d5 R s 2 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 The rst transformation is as follows De ne 9 92 I log 2 7 zdu I log 2 7 zwgzdz which is taken to be analytic in C 70071 Using 927 we de ne a new matrix valued function the new unknown Bz7 as follows 10 132 e WAZENgz7 ug We will verify that B satis es a new RiemanniHilbert problem RiemannHilbert Problem 2 Find a 2 X 2 matrix Bz Bz nN with the properties Analyticity Bz is analytic for 2 E C R and takes continuous boundary values Bz Bz as 2 tends to z with z E R andz 6 3 2 6 C7 Jump Condition The boundary values are connected by the relation 7Nltgltxgtegiltxgtgt eNltgltxgtgiltxgtevltxgtei 11gt Bltzgt Bio 0 eNg 1797 05 Normalization The matrix Bz is normalized at 2 00 as follows 12 lim Bz ll zaoo Here is a very useful result concerning the function g de ned7 as used7 above 2 on 31 Therceusl u 5 gt 0 mm form n e m 1 mnnwmg hunk m g h mm and g 7 mum men on m E M m my 2 pm 1 Myquot gunman u c I L n and n a may an m 2m m WWW amuwsms mm mm a ma The luncunn g 7 3 WWW an anmylic mnlinualinn G to m xtnps 5 Re 1 lt 1 um lt m such am Re Gmgtn inrzCF ySi am 11 Gmlta runswmm 3 In an 4 m 71Mrr mm m i 5pm wlrx xk rf mu REFERENCE m A mm A m mA V mum quotDam Pandev 31mm and that We mm mmquot comm Mam m m mm mm
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