Asymptotics of Hermite polynomials

Transcription

1 Asymptotics of Hermite polynomials Michael Lindsey We motivate the study of the asymptotics of Hermite polynomials via their appearance in the analysis of the Gaussian Unitary Ensemble (GUE). Following Tao [3], we prove various facts about the Hermite polynomials and analyze their asymptotics via analysis of the semiclassical harmonic oscillator operator, which arises from the Hermite differential equation. Still following Tao [3], we use these asymptotics to prove a semicircular law for the expected empirical spectral distribution of the GUE. Since this project is a pedagogical exercise, we provide significantly more detail than Tao [3], filling in several computations and exercises left to the reader, and even correcting a few errors. Lastly, we make use of our results on the asymptotics of Hermite polynomials to show that, in a certain sense, the normalized stationary states for the quantum harmonic oscillator converge to the classical position distribution for the harmonic oscillator. This convergence is suggested graphically in Griffiths []. Our proof is very likely nothing new, though it has been thought out independently by the author. Before beginning, we note that our discussion of random matrix theory (focusing on the GUE) is less detailed than the material that follows (i.e., the discussion of Hermite polynomials, their asymptotics, and the proofs of the semicircular law and the fact about harmonic oscillator), and we refer to other texts for proofs in certain places. (Since this project is for a class on orthogonal polynomials, we decided that it would be more appropriate to provide full detail in the latter parts and to include the discussion of random matrix theory mainly as motivation.) Contents The GUE and related ensembles The Ginibre formula 3 Enter orthogonal polynomials 3 Hermite polynomials 5. The Hermite recurrence relation Differential equations associated with Hermite polynomials Rewriting K n in terms of Hermite functions Analysis of the eigenfunction equation of L h 6 Asymptotics for the GUE expected empirical spectral measure

2 7 Connection with the quantum harmonic oscillator 6 The GUE and related ensembles The n-dimensional GUE is an ensemble of n n Hermitian matrices. A sample M is obtained by sampling the upper triangular entries from the N C (, ) distribution and sampling the diagonal entries from N R (, ), allmutuallyindependently,andfixingthelower-triangularentriesbytheconstraint that M is Hermitian. We write M (M ij ). Consider the measure dm Q n i dm Q ii j<k dm jk < dm jk on the space M of Hermitian matrices (simply the Lebesgue measure on M considered in the most obvious way as R n ). t is clear by construction that the probability measure corresponding to the GUE is given by! Y n µ n C n e Mii / dm Y e Mij dm jkdm < jk A i C n n Y i e Mii / C n e tr(m )/ Y applej<kapplen applej<kapplen e Mij A dm where C n is chosen so that µ n (M). t is clear that e tr(m )/ is invariant under the transformation M 7! UMU for unitary matrices U. By computing the determinant of M 7! UMU as a linear transformation R n! R n (as in Deift []), it can be shown that dm is also unvariant under this transformation, and therefore µ n is invariant under conjugation by unitary matrices. More generally we can consider the distribution µ F n Cn F e F (M) dm, where F : M!Ris invariant under unitary conjugation (so F depends symmetrically on all of the eigenvalues). Particularly noteworthy are the functions F tr P, where P is a polynomial of even degree with positive coefficient for the highest-order term (ensuring integrability). Many of the properties of the GUE hold for these ensembles as well. For P quadratic, it is easy to see that e trp (M) splits into a product of functions of M ii and M jk (with j<k)individually,soevidentlythediagonalandupper triangular entries of M are all mutually independent in this case (as in the GUE). As mentioned by Deift [], a suitable converse is true, and it follows that modulo shifting by a constant and rescaling, the GUE is the unique Hermitian matrix ensemble that is invariant under unitary conjugation with all non-lower-triangular entries mutually independent and therefore a very natural object to study. The Ginibre formula We consider M to be the random matrix variable associated with µ F n (though we shall omit the F from our notation) for F tr P as above. Consider the map which maps M 7! (,,, n) where apple apple apple n are the ordered eigenvalues of M (all real since M is hermitian). Then the following holds for the density n for on the space R n apple {x apple x apple applex n }:

3 Theorem. (Ginibre formula). where n ( ) const e n( ) Y P n i P ( i) n( ), applei<japplen ( j i ). We give a brief sketch of the proof, following Deift []. First note that when the eigenvalues of M are simple, the diagonalization M U U (with the entries of ordered) is unique up to right-multiplication of U by diagonal unitary matrices (which form the maximal torus T n U n ). This holds because a matrix can only commute with if it is diagonal. Also, it is not hard to see that generically M has simple eigenvalues (this can be shown for example by considering the vanishing set of the discriminant, which does not have full dimension). Then there is a map M 7! (R n < { < < < n }, U n mod (T n )) which is a bijection when restricted a subset of M of full measure. AcalculationusinglocalcoordinatesofU n mod (T n ) shows that the Jacobian determinant of is equal to the product of n( ) and a factor that depends only on unitary variables associated with the coordinates on U n mod (T n ), and with some work the Ginibre formula follows from this fact. 3 Enter orthogonal polynomials The GUE and related ensembles are linked to orthogonal polynomials via the expression n( ) that appears in the Ginibre formula. n( ) is of course the Vandermonde determinant det V where V ( i j ). For all j, let j be a monic polynomial of degree j. Then by elementary row operations, we have that n( )det(v ) where V i ( j ),so! nx n( ) det V > V det k ( i ) k ( j ). Then in the case of the GUE, n ( ) const e P n i const det i / det k nx k ( i )e k! nx k ( i ) k ( j ) k const det (K n ( i, j)) applei,japplen, i / k ( j )e applei,japplen applei,japplen! j / applei,japplen where K n is defined by K n (x, y) P n k k( i )e i / k ( j )e j /. We see that taking the j to be non-monic only changes the formula for n ( ) by a constant factor, so let us take j to be the orthonormal Hermite polynomials. Then evidently by orthonormality we have that R K n (x, x) dx n (trace identity) and K n (x, y) R K n (x, z)k n (z,y) dz (reproducing formula). These two properties yield the following: 3

4 Lemma. (Determinantal integration). det (K n ( i, j)) applei,japplek+ d k+ (n k) det (K n ( i, j)) applei,japplek. n particular it follows that det (K n ( i, j)) applei,japplen d d n n! Proof. See, for example, Tao [3]. The latter statement can be used to show that in fact n ( )det(k n ( i, j)) applei,japplen in the GUE case. Furthermore, it can be shown that k (,..., k) :det(k n ( i, j)) applei,japplek is the k-point correlation function for the spectrum, i.e., 3 X k (,..., k)f (,..., k) d d k E F ( i (M n ),..., i k (M n )) 5 R k applei,...,i k applen, distinct for any measurable F : R k! C supported on {x apple applex k }.SeeTao[3]fordetails. t follows from the Christoffel-Darboux formula that K n (x, y) n(x) n (y) n (x) n (y) e (x +y )/, a n (x y) where a i are defined by the three-term recurrence i+ (x) (a i x + b i )P i (x) c i P i (x). Then also we have that K n (x, x) (a n ) n(x) n (x) n (x) n (x) e x /. Then it is clear that pinning down the asymptotics of the n (and the a n ) will help us to grasp the asymptotic behavior of the expected spectral measure of the GUE. Many objects associated with the GUE can be expressed conveniently in terms of K n ;see,for example, Deift []. We will later focus on a particular object: the expected empirical spectral measure of M n. The empirical spectral measure µ Mn is defined by µ Mn P n i i(m n). Recalling our result on the k-point correlation function above with k,wehavethat nx K n (x, x)f (x) dx E F ( i (M n )) for measurable F : R! C. Ofcourse, nx F ( i (M n )) F R i i nx i i(m n), so it follows that Eµ Mn K n (x, x) dx. As we shall confirm later, the range of the eigenvalues of M n scales as p n,soforstudyingasymptoticbehavioritmakessensetoconsidertherescaled probability measure n / K n ( p nx, p nx) dx, which is easily seen to be equal to Eµ p Mn/ n. We write E n n / K n ( p nx, p nx), soeµ p Mn/ n E n (x) dx. We will examine below the asymptotics of E n and Eµ p Mn/ n.

5 Hermite polynomials Henceforth we denote by P i the Hermite polynomials with positive leading coefficients which are orthonormal with respect to the weight function w(x) e x /.. The Hermite recurrence relation By the three-term recurrence, there exist a i,b i,c i (with a i 6, c )suchthat P i+ (x) (a i x + b i )P i (x) c i P i (x). We aim to compute the a i,b i,c i. By taking the inner product of both sides with P i+ w and P i w, we obtain a i xp i (x)p i+ (x)w(x) dx (.) c i a i xp i (x)p i (x)w(x) dx. (.) From (.) and (.) we see that c i a i a i for all i. Meanwhile, taking the inner product of both sides of the recurrence relation with P i w yields b i a i xp i (x) w(x) dx. (.3) (Thus far we have used only the orthonormality of the P i.) Notice that xw(x) w (x). Thus we obtain by integrating by parts in (.): a i P i (x)p i+ (x)+p i (x)pi+(x) w(x) dx P i (x)pi+(x)w(x) dx, where the second equality follows from the fact that P i has degree at most i (and hence is orthogonal to P i+ with respect to w). Let i be the leading coefficient of P i. Then Pi+ of order at most i. Then a i P i (x) i+ (i + ) i P i + Q, where Q is a polynomial apple (i + ) i+ i P i (x)+q(x) w(x) dx (i + ) i+ i. t is also clear from the three-term recurrence relation that i+ a i i,soa i a i p, i+ i p. i + i + (i + )a i,i.e., 5

6 t is easy to see that P (x) ( ) /,so i Also, since c i a i a i for i, wehavethat for i. c i /p i!. p i p i + Now integrating (.3) by parts, we have b i a i P i (x)p i (x)w(x) dx, where the second equality follows from the fact that P i has degree at most i. Then the Hermite recurrence relation is P i+ (x) p i + xp i (x) p i p i + P i (x). (.). Differential equations associated with Hermite polynomials Next we obtain differential equations satisfied by the Hermite polynomials. To do so, we write Pi+ as a linear combination of Hermite polynomials of order at most i, obtainingthecoefficientsofthis linear combination by taking inner products. Note that by integration by parts we have P j (x)pi+(x)w(x) dx xp j (x)p i+ (x)w(x) dx Pj(x)P i+ (x)w(x) dx xpj (x) P j(x) P i+ (x)w(x) dx. When j<i, clearly the last integral is zero, and when j i, wehavethatthecontributionofthe Pj term is zero, so P i (x)pi+(x)w(x) dx xp i (x)p i+ (x)w(x) dx a i, where the second equality follows from (.). Thus we can write P i+ a i P i p i +P i. (.5) 6

7 Then it follows that h (wp i ) (x) w (x)p i (x)+w(x)pi (x) xp i (x)+ p ip i i w(x). By (.), the last expression is equal to p i +Pi+ (x)w(x), sowehavethat or perhaps more familiarly, Since P ( ) /,wehavethat (wp i ) p i +wp i+, d e x / P i (x) p i +e x / P i+ (x). dx P n (x) ( )n ( ) /p n! ex / dn dx n e x /. (.6) This gives an explicit formula for the Hermite polynomials. However, we are interested in their asymptotic behavior for large n, which is not obvious from this formula. We need another approach to uncover this asymptotic behavior. Note that by (.) and (.5), we have for n that np n (x) x p p p np n (x) n n Pn (x) xpn(x) Pn (x). Then we have the Hermite differential equation: P n (x) xp n(x)+np n (x). We can in fact derive a differential equation satsified by the n-th Hermite function n P n w /. We simply differentiate twice: n Pn w / +Pn w / + Pn w /. Noting that w / (x) x w(x)/,sotherefore we obtain w / (x) x w(x) /, x n(x) Pn (x)w(x) / xpn(x)w(x) / + 7 P n (x)w(x) /

8 apple Pn (x) apple apple x xpn(x)+ x np n (x)+ n + + x n(x), P n (x) w(x) / P n (x) w(x) / i.e., n(x) apple n + x n(x). (.7) Note that, equivalently, we have that L n n + n, where we denote by L the harmonic oscillator operator L (x) (x)+ x..3 Rewriting K n in terms of Hermite functions Recall from above that (following from the Christoffel-Darboux formula) K n (x, x) a n P n(x)p n (x) P n (x)p n (x) w(x). Since P n a n P n (see (.5)), it follows that K n (x, x) P n(x) P n (x)p n (x) w(x). (.8) Now so Then n(x) + n n(x) P n(x)w(x) / x P n(x)w(x) /, n(x) P n(x) w(x) x xp n (x)p n(x)w(x)+ x P n(x) w(x) n(x) P n(x) w(x) xp n (x)p n(x)w(x)+np n (x) w(x) P n(x) w(x)+[ xp n(x)+np n (x)] P n (x)w(x) P n(x) w(x)+ x p np n (x)+np n (x) P n (x)w(x), 8

9 where the last equality follows from (.5). By the Hermite recurrence relation (.), x p np n (x)+np n (x) p n(n )P n (x), and again by (.5), the RHS of this equation is equal to Pn (x), so n(x) + n Then by (.8), we have that x n(x) P n(x) P n (x)p n (x) w(x). K n (x, x) n(x) + n x n(x). Recall from above that we were interested in the asymptotic behavior of E n n / K n ( p nx, p nx), since the rescaled expected empirical spectral measure of the GUE is given by E n dx. Now E n n / n( p nx) + p x n n( p nx) n n(x) + x n(x), where n is the rescaled Hermite function defined by n (x) :n / Notice that n satisfies the differential equation n n(x)+ x n(x) n 3/ n( p / x nx)+n n / n apple n + apple n + n + n x n, n / n ( p nx) + n x n( p nx). n( p / x nx)+n n ( p nx) n( p nx)+ x n/ n( p nx) i.e., L /n n + n n, where L h denotes the semiclassical harmonic oscillator operator L h h + x. Then it is natural to analyze the eigenfunction equation or equivalently, L h, x h k(x), (.9) where k(x) : p x / (so we consider only the region where x / < ). We will eventually take h> to be /n (small when n is large) and to be + n (nearly when n is large). 9

10 5 Analysis of the eigenfunction equation of L h t is helpful, both intuitively and formally, to draw a comparison between (.9) and the familiar ODE h k with constant coefficient, having as its solution (x) Ae ikx/h + Be ikx/h. We then forward the ansatz (x) A(x)e i (x)/h + B(x)e i (x)/h, where (x) R x k(y) dy, i.e., is an antiderivative of k. Weobtainbydifferentiating: (x) ik(x) h A(x)e i (x)/h B(x)e i (x)/h + A (x)e i (x)/h + B (x)e i (x)/h. Since we are solving for two functions A and B, heuristicallyweexpecttobeabletoputan additional constraint on A and B, namely A (x)e i (x)/h + B (x)e i (x)/h. (5.) We will see later that the expressions for A and B that we obtain do indeed satisfy this equation. Again we differentiate, yielding (x) Referring back to (.9), we see that then k(x) h (x)+ ik (x) A(x)e i (x)/h B(x)e i (x)/h h + ik(x) A (x)e i (x)/h B (x)e i (x)/h. h A (x)e i (x)/h B (x)e i (x)/h k (x) A(x)e i (x)/h B(x)e i (x)/h. k(x) Adding and subtracting this equation with (5.), we obtain or equivalently A (x)e i (x)/h k (x) A(x)e i (x)/h B(x)e i (x)/h k(x) B (x)e i (x)/h k (x) A(x)e i (x)/h B(x)e i (x)/h, k(x) A (x) B (x) k (x) k(x) A(x)+ k (x) i (x)/h B(x)e k(x) k (x) k(x) B(x)+ k (x) k(x) A(x)ei (x)/h. Define a(x) k(x) / A(x) and b(x) k(x) / B(x), so A(x) k(x) / a(x), B(x) k(x) / b(x),

11 and also i.e., and similarly a (x) k(x) / A (x)+ k (x) A(x) k(x) / k (x) i (x)/h B(x)e k(x) / k (x) k(x) b(x)e i (x)/h, a (x) k (x) k(x) b(x)e i (x)/h, (5.) b (x) k (x) k(x) a(x)ei (x)/h. Then on any compact interval contained within (, ), wehavethata,b O( a, b ). Furthermore, by Gronwall s inequality we have that a(x) applec b() and b(x) applec a() on for constants C,C depending only on. Therefore on. Now by (5.), we have a(x) a() x x a,b,a,b O( a() + b() ) (x) a (u) du k (u) k(u) b(u)e i (u)/h du k (v) k ( (v)) (v)b (v) e iv/h dv, where we have made the substitution v (u) (note that is strictly increasing, hence invertible, since k is positive). Then integrating by parts we obtain " # v (x) a(x) a() h (v) (x) i k ( (v)b (v) e iv/h F (v)e iv/h dva, (v)) where " F (v) d k (v) dv k ( (v)) (v)b n particular is a smooth function on that does not depend on h, and furthermore from the product rule it is clear that F O( b + b ). t follows from our earlier finding that F O( b() + b () ). Evidently then a(x) a() + O (h [ a() + b() ]), v (v) #.

12 and by analogous reasoning we obtain We have thus shown the following: b(x) b() + O (h [ a() + b() ]). Theorem. Suppose that L h. Then on any compact interval contained in (, ), where Furthermore (x) k(x) / a(x)e i (x)/h + b(x)e i (x)/h, a(x) a() + O (h [ a() + b() ]), b(x) b() + O (h [ a() + b() ]). (x) ik(x)/ h a(x)e i (x)/h b(x)e i (x)/h. 6 Asymptotics for the GUE expected empirical spectral measure Now plugging in + n and h n,weseethatonanycompactinterval contained in (, ), where and furthermore n k(x) / a(x)e in (x) + b(x)e in (x), a(x) a() + O n [ a() + b() ], b(x) b() + O n [ a() + b() ], n(x) nik(x) / a(x)e in (x) b(x)e in (x). n the following the reader should note that a and b depend on n, thoughweomitthisdependence from the notation. Recall from above that the density of the expected empirical spectral measure is given by so E n (x) x n (x) + n n(x), E n (x) a(x)e in (x) + b(x)e in (x) k(x) x + a(x)e in (x) b(x)e in (x) k(x) h a(x) + b(x) +< a(x)b(x)e in (x) i k(x) h + a(x) + b(x) < a(x)b(x)e in (x) i k(x) a(x) + b(x) apple k(x) x + k(x) x

13 +< a(x)b(x)e in (x) apple k(x) q Notice that k(x) + x n,sok(x) notion more precise. ndeed, we have that applek(x) x x x + n + n + n x x + O x q x n + n x k(x). (6.) for large n. We will make this x x x n. n x + n n + n Notice that for >, wehavethat apple +,and( ) apple. Thus in fact for all, > we have that ( ) apple,so and "k(x) k(x) Similarly, it is easy to show that k(x) x E n (x) a(x) + b(x) h p r x r # x O, n x + o(). q x + o(). Then we have from (6.) that x i x + o() + O( a(x) b(x) ) o(). Of course since a, b O( a(), b() ), weinfacthavethat E n (x) a(x) + b(x) h p i x + o() + O( a() b() ) o(). (6.) To proceed we approximate the initial conditions a() and b(). Recall(.6) froma above, i.e., that P n (x) ( )n ( ) /p n! ex / dn dx n e x /. Using the Taylor expansion X e x / ( ) k x k k, k! k 3

14 we see that for even n, and for odd n, P n () ( ) n/p n! ( ) / n/ (n/)!, P n(), P n (), P n() ( )(n+)/ (n + ) p n! ( ) / (n+)/ ((n + )/)!. Now n () n / P n () and n() n 3/ n() n 3/ Pn(), andfurthermore n () a() + b() and n() ni(a() b()), so 8 < a() + b() ( )n/ n /p n!, a() b(), n is even ( ) / n/ (n/)! : a() + b(), a() b() ( )(n+)/ n / (n+) p n! i( ) / (n+)/ ((n+)/)!, Now recalling that n! p n (n/e) n (Stirling s approximation), we have that n is odd. ( ) n/ n /p n! ( ) / n/ (n/)! ( ) n/ n/ ( ) / n / n n/ e n/ ( ) / n/ (n/)! ( ) n/ n/ n n/ e n/ n/ (n/)! ( ) n/ n / n n/ e n/ n/ / n / n n/ n/ e n/ ( ) n/ /, so as n!.wealsocompute ( ) (n+)/ n / (n + ) p n! ( ) / (n+)/ ((n + )/)! n /p n! ( ) / n/ (n/)!! / ( ) (n+)/ n / (n + )( ) / n / n n/ e n/ ( ) / (n+)/ ((n + )/)! ( ) (n+)/ (n + )nn/ e n/ (n+)/ ((n + )/)! ( ) (n+)/ (n + )n n/ e n/ (n+)/ / (n + ) / (n + ) (n+)/ (e) (n+)/ n/ n ( ) (n+)/ / e / n + ( ) (n+)/ / e / + n / n ( ) (n+)/ /, so n / (n + ) p n! ( ) / (n+)/ ((n + )/)!! /.

15 Now when n is even, a() b(), soa() b() (a() + b()). Whenn is odd, a() b(), so a() b() (a() b()). t follows that we have that a(), b()! / as n!,so a() + b()! as n!. Recall (6.) from above, i.e., that E n (x) a(x) + b(x) h p i x + o() + O( a() b() ) o(). Also recall that a(x) a() + O n [ a() + b() ], b(x) b() + O n [ a() + b() ]. But since a(), b()! /,so a() + b() and a() b() are both bounded uniformly in n, and in fact a(x) a() + O(n ), b(x) b() + O(n ), and O( a() b() ) O(). Thus a + b a() + b()! uniformly on as n!, so a + b! uniformly on as n!,i.e., a + b + o(). Then we have shown that apple hp i E n (x) + o() x + o() + o() p x + o(), or, in other words, E n (x) p x converges to zero uniformly on. Now let : R! R be measurable, bounded, and supported on a compact interval (, ). Now E n (x) p x! uniformly on, so (x) E n (x) p x dx! by bounded convergence, and consequently R E R n! (x)p x dx. n particular, letting m [ +m, m ] and taking m we see that R m E n! R p m x dx. Ofcourse, R R E n, so it follows that R R p R\ m E n! m x dx. Now let : R! R be bounded and continuous, so appleb everywhere for some B>. Then q (x) E n (x) ( x p ) + dx apple (x) E n (x) x dx m + (x) E n (x) dx R\ m + (x) q ( x R\ m ) + dx. 5

16 The first term in the last expression converges to zero by our previous argument, and the second term is bounded by B R R p R\ m E n (x) dx, which converges to B m x dx. Thus lim sup n! (x) E n (x) q ( x ) + dx apple B p m x dx. The RHS p of this inequality can be made arbitrarily small by taking m sufficiently large since x dx,soinfact R t follows that lim n! (x) E n (x) q ( x ) + dx. E n! q (x) ( x ) + dx. We collect our results into the following theorem: q Theorem. Eµ p Mn/ n E n dx! sc dx weakly, where sc (x) ( x ) +. Furthermore, E n! sc uniformly on any compact interval contained in (, ), and E n! sc in L on R\[, ]. 7 Connection with the quantum harmonic oscillator Recall from the preceding section that we denote by P n the Hermite polynomials orthonormal with respect to w(x) e x /.Furtherwehavedefined n P n w / and n (x) n / n( p nx), so n p n n( p nx) dx n dx P n (x) w(x) dx, i.e., k n k L. n fact the n are (modulo horizontal rescaling) the normalized stationary states for the quantum-mechanical harmonic oscillator. See, for example, Griffiths [] for a discussion of the harmonic oscillator. n Griffiths [], it is indicated graphically that the n converge in some sense to the density of the classical distribution of the harmonic oscillator. We will formulate and prove this convergence precisely. Consider the harmonic oscillator defined by x(t) A cos (!t). t is clear that the amount of time per period for which x(t) y (for y [ A, A]) isgivenby! cos (y/a), andtheperiodofoscillationis /!, so the fraction of time for which x(t) y is given by cos (y/a). 6

17 t follows the stationary distribution function F forthe position of the harmonic oscillator satisfies F (y) cos (y/a) for y [ A, A], andthecorrespondingprobabilitydensityfunctionis f(y) F (y) A x A /. Without loss of much generality, we fix A,so f(x) x /. + n fact we must rescale the have the following: n to obtain convergence to the classical density f. Moreprecisely,we Theorem. For any bounded continuous function : R! R, R n! R f. Proof. Let : R! R be continuous with appleb for some B>. From the preceding section, we have that on any compact interval contained in (, ), n(x) k(x) a(x)e in (x) + b(x)e in (x), where a(x) a() + O n, b(x) b() + O n. (Recall that a and b depend on n). Furthermore, we showed that a()! / and b() ( ) n a(). Then we have that on. n(x) k(x) a() e in (x) +( ) n e in (x) + O n Let for x ( g n (x) k(x) a() e in (x) +( ) n e in (x), ). Weclaimthat R R n g n!. ndeed, and a() is bounded uniformly in n, so by bounded convergence, and the claim follows. n(x) g n (x) k(x) a() O(n ), n g n! Then define h n (x) k(x) e in (x) +( ) n e in (x) 7

18 for x (, ). Weclaimthat R g R n h n!. ndeed we have that g n h n k(x) e in (x) +( ) n e in (x) a(), so again our claim follows from bounded convergence. Now h n k(x) e in (x) +( k () e inx +( (x) ) n e in (x) (x) dx e inx +( ) n e inx ) n e inx (x) dx, (x) (x) dx where (x) k (x) (x) (x) ()(x), so L (R) \ L (R). Then by the Plancherel identity, since h F e inx +( ) n e inx i (u) ( ) u n n + (u)+( ) n u + n, we have that h n apple ( ) n ˆ By the Riemann-Lebesgue lemma, ( ) n hˆ n Thus as n!. ˆ() k () (x) k(x) (x) dx. h n n + ˆ + ˆ n (x) k(x) (x) dx! n +ˆ(). i! as n!.furthermore, (x) dx / x We showed in the preceding section that k(x)! uniformly on as n!,and / since k is bounded away from on it follows that k(x)! f(x) uniformly on. Therefore k(x) (x) dx x f dx! 8

19 as n!. Putting together our results, we have shown that R n! R f dx as n!. Now we could have taken in the preceding, so in particular we have that R n! R f as n!.ofcourse, n is a probability density, so R R R\ n! f. Now R n R f apple n apple n! B f f + R\ f + B. R\ n + n + B R\ R\ f f Now by taking [ +m, m ] for m sufficiently large, we can make the last expression above arbitrarily small (since R f ). Therefore R R n! R f as n!,andtheproofis R complete. References [] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, American Mathematical Society, 999. [] D. J. Griffiths, ntroduction to Quantum Mechanics, Pearson, 5. [3] T. Tao, Topics in Random Matrix Theory, AmericanMathematicalSociety,. 9