Class Note for PHYS 142 at UA
Popular in Course
Popular in Department
This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 17 views.
Reviews for Class Note for PHYS 142 at UA
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 02/06/15
wig mg wig mgnn u m a w w I me o E ni ngs em fmai aa s as 2 a 5quot 23 a a if a 155 3 22 3 5 55 EEEE EEE rg gaaags gs gi E s anar m agsizam g s 35 52 555232 i1 is 2 55 is E12 awe w 54 s m 23 a a in a 51 23 g 555 92 253 SEE 55 ea auda ax m 555 139 Eu 7 r39 411 21quot 4 x cm mewgmg garcsmz I 0 v sin1vrj39llnlt4 d9 7 urcsmz u Fox 26 omwe have wwwe f mgtgtltlo Fox 26 01mm have 1 AimZ f zn r 7 M 1 l J 1 4Arm mquotl 1 0 quot formuluil lsw holds r3 L formula mm holds if a I lJ nI is replaced hy 1l ngt quot Lformuln 1mm hnlds imo mm is replaced by 00 u lngnl c For n 3 lt I there exlsl Chivalued funminns 02 m see lt730gt for a delinilionl uniformly bounded in n and 7 such that ma w lrzr 39 n3 quot m x lmm iwvaay vLd5u1j m z In M lilmmdy m my and rgmm zl are bounded by 04391quot Uh quot 939 for any u Mann WWW v 5quot The error tenns rumquot c a r EQUILIBRIUM MEASURES The equilibrium measure is de ned as the unique minimizer in 1 M1R Probability measures on R of the functional lt2 Ii M a 700001 i H A loglz 7 w d1dy A WWI In the lecture we will discuss the various origins of this variational problem and how it relates to orthogonal polynomials and random matrix theory For now the main points are these 0 The equilibrium measure itquot is ac wiriti Lebesgue measure and its density is exactly ibg appearing in the asymptotic formulae of the previous section 1 u il 3 z 7 7d ltgt 1am 7 M o The function is also the limiting mean density of eigenvalues More precisely if we consider random Hermitian matrices with probability measure of the form u z 6 711 4 LeinsTr lMl dM ZN note the N appearing in the exponential above then we have the following limit theorem N s ngnm A gtltzgt was Recall from the previous lecture that we have following RiemannHilbert problem which is known to characterize the polynomials pgN 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 orthogonal with respect to e NW l I 1 eiNVw lt6 Altzgt A7ltzgt 0 l Normalization The matrix Az is normalized at 2 00 as follows 2 0 7 lim Az lli zaoo 0 Zn The connection between these orthogonal polynomials and the solution of RiemannHilbert Problem 1 is the following 1 1 pnsgte NVlt5gt mm 2mm 8 7 2 d5 lt8 Altzgt 72Wis N 7 N Pn71se NV5 d8 n71n71pn1z nilmil R 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 REFERENCES 1 A Fokas A lts and A V Kitaev Discrete Painleve equations and their appearance in quantum gravity Commun Math Phys 142 3137344 1991
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'