Deift/Zhou Steepest descent, Part I

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1 Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n, tc... Th dpndnc on t R will b omittd in th notation. 9. Wak asymptotics Writ π n.n = n j= (z η j,n), so that η j,n ar th zros of π n,n. W will show that for t 2 ths zros accumulat on a singl intrval [ a, a] and hav dnsity dµ V (x) = x2 (c + π 2 ) a 2 x 2, on [ a, a], whr a = c = t+ t 2 /4+3 2t+2 t On can show that with ths constant, th total mass of th masur is, i.. a a dµ V (x) =.

2 2 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I For intrprting th following thory w not that π n,n(z) = n π n,n n n j= = z η j,n z x dµ n(x). whr µ = n n j= δ η j,n is th normalizd sum of dirac masss at th zros of π n,n. Thorm 9... Lt t 2. Thn π n,n(z) lim = n n π n,n a a z x dµ V (x), uniformly for z in compact substs of C \ [ a, a]. W will prov this Thorm, among strongr rsults, by using Rimann- Hilbrt tchniqus. In this lctur w will always assum Th cas t = 2 will b don latr. 9.2 Th g-function t > 2. Dfinition W dfin g : C \ (, a] as g(z) = a a log(z x)dµ V (x), whr w choos th branch of th logarithm such that g is analytic and ral for z > a. Lmma g (z) = V (z) 2 (c + z 2 /2)(z 2 a 2 ) /2, whr tak th z (z 2 a 2 ) /2 such that it is analytic in C \ [ a, a] and positiv on (a, ). Proof. Dfin th function h = Cauchy transform of dµ V (x), i.. h(z) = 2i 2i g. a a Thn h is, is up to constant, th x z dµ V (x).

3 9.2. THE G-FUNCTION 3 But that mans that h is th uniqu solution to th following RHP (xrcis!) h is analytic in C \ [ a, a] h + (x) h (x) = (c + x2 2 ) a 2 x 2, x ( a, a), h(z) = O(/z), z h is boundd nar ± a. W claim that 2i (V (z)/2 (c + z 2 /2)(z 2 a 2 ) /2) solvs th sam RHP problm and hnc,by uniqunss of th solution, w provd th statmnt. To s that this indd solv th RHP w nd to chck th jump condition, which follows by th choic of th squar root (xrcis), and th asymptotic bhavior at infinity. Aftr a computation on chcks that V (z) cancls th polynomials part of th scond trm and hnc th claim follows. Th following proprtis will b crucial in th upcoming analysis. Lmma Th function g satisfis th following proprtis g + (x) g (x) = 2πi for x < a g + (x) g (x) = 2πi a x for x [ a, a]. g + (x) + g (x) V = l for x [ a, a] g + (x) + g (x) V < l for x R \ [ a, a]. for som constant l R. Proof.. Not that For z, x R w hav log ± (z x) = log z x + i arg ± (z x) arg + (z x) arg (z x) = Sinc dµ V =, w find th statmnt. 2. Follows by th sam argumnts as. { 2πi, z < x 0, z x.

4 4 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I 3. This follows form th prvious lmma. By th dfinition of th squar root w hav (z 2 a 2 ) /2 ± = ±i a 2 z 2 for z ( a, a) and hnc g +(x) + g (x) V (x) = 0 for x [ a, a]. 4. This follows form th sam argumnts as bfor but now w hav g +(x) + g (x) V (x) < 0 on (a, ) on (, a). g +(x) + g (x) V (x) > Ovrviw of th Dift/Zhou stpst dscnt analysis W start with th RHP for th orthogonal polynomials as discussd int h prvious lctur (with w(x) = nv (x) ). RH problm W sk for a function Y : C \ R C 2 2 such that Y is analytic in C \ R. ( ) nv (x) Y + (x) = Y (x), for x R. 0 ( ) z n 0 Y (z) = (I + o()) 0 z n as z. Th asymptotic analytis consists of a squnc of transformations Y T S R. Each transformation is simpl and invrtibl. Aftr ach transformation w obtain a nw RiHP. In th nd, th goal is to nd up with a RHP of th form { R + = R J r R(z) = I + O(/z), z, with J R as n. W thn find R I and morovr, an asymptotic xpansion for R. By tracing th transformations back, w obtain an asymptotic xpansion for Y.

5 9.4. FIRST TRANSFORMATION : NORMALIZING THE RHP NEAR INFINITY5 9.4 First transformation : normalizing th RHP nar infinity In th first transformation ( ) ( ) nl/2 0 n(g l/2) 0 T (z) = 0 nl/2 Y (z) 0 n(g l/2). Thn T satisfis RHP that w will now pos. First of all, w not that sinc µ V has total mass w hav g(z) = log z + O(/z), z. That mans that T (z) = I + O(/z), which mans that w hav normalizd th RHP at infinity. Of cours, th pric w pay is in trms of a mor complicatd jump structur which w can comput from ( ) ( ) J T = T T n(g l/2) 0 + = 0 n(g Y l/2) Y n(g l/2) n(g l/2). A straightforward computation thn givs that T solvs th following RHP. RH problm W sk for a function T : C \ R C 2 2 such that T is analytic in C \ R. ( n(g + (x) g (x)) T + (x) = T (x) n(g ) +(x)+g (x) V (x) l) 0 n(g +(x) g (x)), for x R. T (z) = I + O(/z) as z. Thr is a simplification for th jump matrix for T by th proprtis of th g-function in Lmma Thn ( n(g + (x) g (x)) n(g ) +(x)+g (x) V (x) l) 0 n(g +(x) g (x)) ( ) n(g +(x) g (x)) 0 n(g, x [ a, a] +(x) g (x)) = ( ) n(g +(x)+g (x) V (x) l), x R \ [ a, a]. 0 (9.4.)

6 6 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I ( ) nϕ a ( nϕ + ) ( ) nϕ 0 0 nϕ a 0 Figur 9.: Jumps for T W will introduc som notation for prsntation purposs. W dfin, for z C \ (, a] ϕ(z) = 2 z a (c + y2 2 )(y2 a 2 ) /2 dy, whr th path of intgration is C \ (, a]. Thn ϕ is analytic. Morovr, ϕ + ϕ = 0 mod 2πi on (, a] and thrfor w hav that nϕ is analytic in C \ [ a, a]. Morovr, by Lmmas and w s that w can rwrit th jump in trms of ϕ and obtain ( ) nϕ + T + = T 0 nϕ on [ a, a] and nϕ ( T + = T 0 ) on R \ [ a, a]. S also Figur??. Sinc R ϕ > 0 on R \ [ a, a] w s that th jumps on R \ [ a, a] ar xponntially small in n. Howvr, th jumps on [ a, a] ar not xponntially small. In fact, ϕ ± (x) = ±2πi a x dµ V (x) and hnc th jumps on [ a, a] ar highly oscillating! 9.5 Scond Transformation: opning of th lnss To liminat th highly oscillating jumps w prform a transformation that is known as th opning of th lns. It is basd on th following factorization ( ) ( ) ( ) ( ) nϕ nϕ = nϕ 0 nϕ +. Now w do th following. W opn up a lns around th intrval [ a, a]. That is, w tak two simpl contours conncting a to +a, on in th uppr

7 9.5. SECOND TRANSFORMATION: OPENING OF THE LENSES 7 half plan and on in th lowr half plan. W thn dfin ( ) 0 T nϕ in uppr part ( ) S = 0. T nϕ in lowr part T lswhr. S also Figur 2.2. Now what did w gain from this transformation. Proposition For t > 2 w can choos th lips of th lns such that R ϕ < 0 on th lipss. Proof. Not that ϕ is analytic in C \ (, a] and, for x [ a, a], ϕ + (x) = 2πi a x dµ V (x). Bcaus of th Cuachy Rimann quations (not that ϕ can analytically continud to [ a, a]) w hav with z = x + iy d R ϕ dy (x) = d Im ϕ dx (x) = dµ V dx (x) > 0 (sinc t > 2). But that mans that w can indd dform th intrval [ a, a] to a path from a to a in th uppr half plan such that R ϕ < 0 on that path. Th argumnt for th lowr lips is analogous. In Figur 2.4 w show th part of th complx plan whr R ϕ < 0 for t =. Clarly, th lns can b chosd such that it falls compltly in th rgion. Conclusion: all jumps for S othr thn th jump on [ a, a] ar xponntially small in n (pointwis). Th jump on [ a, a] dos not dpnd on n. Hnc w hav mad big progrss in th undrstanding of th asymptotic bhavior of Y.

8 8 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I S = T S = T S = T ( ) 0 nϕ ( ) 0 nϕ Figur 9.2: Opning of th lns ( ) 0 nϕ ( ) nϕ 0 ( 0 ) 0 ( ) nϕ 0 ( ) 0 nϕ Figur 9.3: Jumps for S

9 9.5. SECOND TRANSFORMATION: OPENING OF THE LENSES Figur 9.4: Th whit rgion in which is th rgion whr R ϕ < 0 for t =. Th shadd rgion is th rgion R ϕ > 0.

10 0 LECTURE 9. DEIFT/ZHOU STEEPEST DESCENT, PART I

11 Bibliography [] M.J. Ablowitz and A.S. Fokas, Complx variabls: introduction and applications. Scond dition. Cambridg Txts in Applid Mathmatics. Cambridg Univrsity Prss, Cambridg, [2] K. F. Clancy and I. Gohbrg, Factorization of matrix functions and singular intgral oprators. Oprator Thory: Advancs and Applications, 3. Birkhusr Vrlag, Basl-Boston, Mass., 98 [3] P.A. Dift,Orthogonal polynomials and random matrics: a Rimann- Hilbrt approach. Courant Lctur Nots in Mathmatics, 3. Nw York Univrsity, Courant Institut of Mathmatical Scincs, Nw York; Amrican Mathmatical Socity, Providnc, RI, 999. viii+273 pp. [4] A.S. Fokas, A. Its, A.A. Kapav and V.Y. Novokshnov, Painlv Transcndnts. Th Rimann-Hilbrt approach. Mathmatical Survys and Monographs, 28. Amrican Mathmatical Socity, Providnc, RI, xii+553 pp. [5] I. Gohbrg and N. Krupnik, On-dimnsional linar singular intgral quations. I. Introduction. Oprator Thory: Advancs and Applications, 53. Birkhusr Vrlag, Basl, 992.

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!