OPSF, Random Matrices and Riemann-Hilbert problems

Transcription

1 OPSF, Random Matrices and Riemann-Hilbert problems School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, ICMAT October, 207

2 Plan of the course lecture : Orthogonal Polynomials and Random Matrices lecture 2: The Riemann-Hilbert problem for orthogonal polynomials lecture 3: Logarithmic potentials and equilibrium measures lecture 4: Asymptotics and universality

3 Plan for this lecture We are interested in the asymptotic behavior of the eigenvalues of Hermitian random matrices M of order N with distribution Z N e NTrV (M) dm. The eigenvalues form a determinantal point process with kernel K N (x, y) = N k=0 γ 2 k,n P k,n(x)p k,n (y), P m,n (x)p n,n (x)e NV (x) dx = 0, m n. We will use the Riemann-Hilbert problem and the Deift-Zhou steepest descent method to obtain the required asymptotic result.

4 The Riemann-Hilbert problem Y = C C 2 2 with properties Y = Y N is analytic in C \ R. 2 jump condition e NV (x) Y + (x) = Y (x), x R. 0 3 behavior near infinity ( Y (z) = K N (x, y) = I + O ( z 2πi(x y) ) ) ( z N ) 0 0 z N, z. 0 Y (y)y (x). 0

5 Asymptotic analysis of the Riemann-Hilbert problem The idea is to transform the Riemann-Hilbert problem for Y = Y N in a number of steps Y T S R to another Riemann-Hilbert problem for R = R N on a family of contours Σ R in the complex plane, but with jumps on this contour which are close to the identity matrix. R is analytic on C \ Σ R. 2 R + (z) = R (z)v N (z) for z Σ R. 3 R(z) = I + O( z ) as z. with jumps for which both in L 2 and L on Σ R. lim V N(z) = I, N

6 Asymptotic analysis of the Riemann-Hilbert problem Theorem The Riemann-Hilbert problem for R has the asymptotic behavior lim R N(z) = I, N uniformly on compact subsets of C \ Σ R. One has R N (z) = I + max( I V N 2, I V N )O( z + ), uniformly on compact subsets of C \ Σ R. Then invert all the transformations R S T Y to get the asymptotic behavior of Y = Y N.

7 First transformation The first transformation is to normalize the Riemann-Hilbert problem at infinity. Define T (z) = e Nl/2 0 e Ng(z) 0 e Nl/2 0 0 e Nl/2 Y (z) 0 e Ng(z) 0 e Nl/2 with g(z) = log(z x) dµ V (x). A shorter notation is T (z) = e Nlσ 3/2 Y (z)e Ng(z)σ 3 e Nlσ 3/2 where σ 3 = 0. 0

8 Riemann-Hilbert problem for T T is analytic in C \ R. 2 For x R ( e N[g +(x) g (x)] e T + (x) = T (x) N[g+(x)+g ) (x) V (x)+l] 0 e N[g+(x) g, (x)] 3 T (z) = I + O( z ) as z.

9 Riemann-Hilbert problem for T Recall that for x R g + (x)+g (x) = 2 log x t dµ V (t) = 2U(x; µ V ) = V (x) l. and 0, if x > b, g + (x) g (x) = 2πi, if x < a, b 2πi dµ V (t), if x (a, b). x

10 Riemann-Hilbert problem for T For x [a, b] = supp(µ V ) with e 2πiNϕ(x) T + (x) = T (x) 0 e 2πiNϕ(x), and for x (, a] [b, ) b x ϕ(x) = dµ V (t) = dµ V (t), x b e N[2U(x;µ V )+V (x) l] T + (x) = T (x). 0 with 2U(x; µ V ) + V (x) l.

11 Second transformation We want to replace the oscillatory jump on [a, b] by an exponentially decreasing jump on a deformed contour. e 2πiNϕ 0 e 2πiNϕ = ( 0 e 2πiNϕ ) e 2πiNϕ. Instead of jumping over [a, b] in one step, we jump in three steps, using three jump matrices in this product.

12 Opening a lens We introduce two new contours connecting a and b in the upper half plane and the lower half plane a b We jump over the lips of the lens with jump matrix ( ) 0 e 2Nφ where φ is the analytic continuation of iϕ, i.e., φ(z) = z b (q(s)) /2 ds, q = c(s a)(s b).

13 Riemann-Hilbert problem for S Inside the lens we only did one or two of the jumps (instead of all three), but outside the lens everything remains the same (including the asymptotic condition). Define S by S(z) = T (z), outside the lens, 0 S(z) = T (z) e 2Nφ, in the upper part of the lens, 0 S(z) = T (z) e 2Nφ, in the lower part of the lens. 0 S + = S on (a, b) S + = S e 2Nφ on the lips of the lens, 3 e 2NΦ S + = S on (, a) and (b, ). 0

14 The Riemann-Hilbert problem for S Property The lens can be opened so that Rφ < 0 on the upper and lower lips of the lens. This follows from the property that d dx i[g +(x) g (x)] > 0, x (a, b), and the Cauchy-Riemann equations, which then say that the derivative in the vertical direction in < 0 on (a, b). On the real line the real part of φ is 0, hence moving away from the real line in the vertical direction gives Rφ < 0.

15 The Riemann-Hilbert problem for S 0 e 2NΦ e 2Nφ e 2NΦ 0 0 a b 0 0 e 2Nφ 0

16 The global parametrix Four of the jump matrices for S converge to I as N. Only the jump matrix on (a, b) survives the limit. bigskip We expect that the main contribution to S is going to be a matrix G that satisfy the following Riemann-Hilbert problem G is analytic in C \ [a, b]. 2 0 G + = G on (a, b). 0 3 G(z) = I + O( z ) as z.

17 The global parametrix The solution of this Riemann-Hilbert problem is known: G(z) = ( i β(z) 0 i i 0 /β(z) i ) with β(z) = z b /4. z a

18 Third transformation Now consider then R has no jump on (a, b). R(z) = S(z)G R still has jumps on the lips of the lens and on (, a) and (b, ), but these jumps converge to I as N. Warning: this convergence is not uniform in the neighborhood of a and b. Hence we cannot conclude yet that lim N R = I. A local analysis near a and b is still needed.

19 Local parametrices We will solve the Riemann-Hilbert problem for S in the neighborhood of a and b exactly, and then match the solution to the global parametrix on the boundary of the neighborhoods. Near b the Riemann-Hilbert problem for the parametrix P b is 0 e 2Nφ 0 e 2NΦ 0 0 b 0 e 2Nφ

20 Local parametrices We make the jumps constant by using P b = P e Nφ 0 b 0 e Nφ The Riemann-Hilbert problem for P b then becomes b 0

21 Local parametrices This Riemann-Hilbert problem corresponds to the Riemann-Hilbert problem for the Airy function. A solution is given by A(z) = ( Ai(z) ω 2π 2 Ai(ω 2 ) z) iai (z) iωai (ω 2, 0 < arg z < 2π/3 z) A(z) = ( ωai(ωz) ω 2π 2 Ai(ω 2 ) z) iω 2 Ai (ωz) iωai (ω 2, 2π/3 < arg z < π z) A(z) = ( ω 2π 2 Ai(ω 2 ) z) ωai(ωz) iωai (ω 2 z) iω 2 Ai, π < arg z < 2π/3 (ωz) A(z) = Ai(z) ωai(ωz) 2π iai (z) iω 2 Ai, 2π/3 < arg z < 0 (ωz) where ω = exp(2πi/3).

22 Local parametrices To match this with the global parametrix on the boundary of the neighborhood, we can multiply on the left by an analytic matrix. The result is that P b (z) = E N (z)a(f N (z)), where and 2 3 f N(z) 3/2 = Nφ(z), E N (z) = ( i (z a) f ) σ3 /4 N(z). 2 i z b

23 The final Riemann-Hilbert problem Now we can define the final matrix R by S(z)G, outside the neighborhoods of a and b, R(z) = S(z)Pa, in the neighborhood of a, S(z)P b, in the neighborhood of b. The jump on (a, b) disappears, and so do the jumps inside the neighborhoods of a and b. The remaining jumps converge to I uniformly (and in L 2 ) on the contour Σ R. lim R(z) = I, N uniformly on compact subsets of C \ Σ R.

24 Undo the transformations Now return to Y by undoing the transformations. S(z)G, outside the neighborhoods of a and b, R(z) = S(z)Pa, in the neighborhood of a, S(z)P b, in the neighborhood of b. T (z), outside the lens, 0 T (z) S(z) = e 2Nφ, in the upper part of the lens, 0 T (z) e 2Nφ, in the lower part of the lens. e Nl/2 0 e Ng(z) 0 e Nl/2 0 T (z) = 0 e Nl/2 Y (z) 0 e Ng(z) 0 e Nl/2

25 Universality: sine kernel in the bulk For x, y (a + δ, b δ) one has ( K N (x, y) = e Nφ +(y) e Nφ+(y)) S e + 2πi(x y) (y)s Nφ +(x) +(x) e Nφ+(x) S+ (y)s +(x) = G+ (y)r (y)r(x)g + (x) ( = G+ (y) I + O( x y ) N ) G + (x) Then K N (x, y) = π(x y) sin(in[φ +(y) φ + (x)]) + O().

26 Universality: sine kernel in the bulk Take x = x u + Nψ V (x ) and y = x v + Nψ V (x ), with ψ V the density of µ V, then ( ( in φ + x v ) ( + φ+ x u ) ) Nψ V (x + ) Nψ V (x ) Hence lim N Nψ V (x ) K ( N x + = πn x v + Nψ V (x ) x u + Nψ V (x ) π(u v). u Nψ V (x ), x v + Nψ V (x ) ψ V (s) ds ) = sin π(u v) π(u v).

27 References P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics 3, Courant Institute, New York, NY, and Amer. Math. Soc., Providence, RI., M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications 98, Cambridge University Press, 2005 (paperback edition 2009) M.L. Mehta, Random Matrices, revised and enlarged second edition, Academic Press, San Diego, CA, 99. E.B. Saff, V. Totik, Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften 36, Springer-Verlag, Berlin, 997. G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Providence, RI, 939; fourth edition 975.