Fredholm Determinants

Transcription

1 Fredholm Determinants Estelle Basor American Institute of Mathematics Palo Alto, California The 6th TIMS-OCAMI-WASEDA Joint International Workshop on Integrable Systems and Mathematical Physics March 2014

2 Preliminaries The theory of Fredholm determinants for integral operators has roots in the theory of ordinary determinants for the familiar n n matrices and the systems of equations associated to them. So let us begin with a little history of the determinants. We suppose that K(x, y) is continuous and we define the operator T on L 2 (a, b) which takes the function f to T(f )(x) = b a K(x, y) f (y) dy. The function K(x, y) is called the kernel of T.

3 Let us also for a moment suppose we are trying to find a function f such that g = f λt(f ) for some given g. One thing we might do is discretize this problem and break up the interval into small segments and try to approximate a solution. Let δ = b a N and choose x p in a partition of [a, b] of mesh size δ. Then using finite sums as an approximation, we can write for p = 1, 2,..., N. g(x p ) f (x p ) λδ N K(x p, x q ) f (x q ) q=1

4 This system will have a solution if λ is not a zero of the determinant D N (λ) defined by: 1 λδk(x 1, x 1 ) λδk(x 1, x 2 ) λδk(x 1, x N ) λδk(x 2, x 1 ) 1 λδk(x 2, x 2 ) λδk(x 2, x N ).... λδk(x N, x 1 ) λδk(x N, x 2 ) 1 λδk(x N, x N )

5 Set ( x1, x K N y 1, y N ) ( ) = det K(x p, y q ), p, q, = 1,, N. Then grouping terms by powers of λ we have D N (λ) = 1 λδ ( xp K )+ (λδ)2 ( xp1 x K p2 x p 2! x p1 x p2 p p 1,p 2 ) + Now let δ 0 and we formally have that D(λ) is ( ) ( ) x1 1 λ K dx x 1 + (λ)2 x1 x K 2 dx 1 2! x 1 x 1 dx 2 + 2

6 What about convergence? To show that this series defines an entire function (at least for a finite interval) we let M be a bound for K(x, y). Note that if a matrix A has columns A 1,, A n then det(a) A 1 A n. This means that each integrand ( ) det K(x i, x j ) has absolute vaue at most M n n n/2 and that each term of the series is at most λ n (b a) n M n n n/2 n! and thus the series converges for all λ.

7 Thus D(λ) is an entire function and it can be shown using the very same techniques just employed that if λ is a zero of D(λ) then 1 is an eigenvalue of the operator T. λ The set of non-zero eigenvalues is discrete and either constitutes a finite set or the eigenvalues have a limit point of zero. Fredholm did more than this in his original paper. He produced the kernel of the inverse of I λt.

8 We suppose D(λ) is not zero, and we define D(x, y, λ) by ( λ) n 0 n! If S is the operator with kernel ( x x1 x K n y x 1 x n D(x, y, λ) D(λ) then (I λt) and (I λs) are inverses. ) dx 1 dx n.

9 In addition to the determinant, we can also define the trace of the integral operator as b a K(x, x) dx. This is motivated by the fact that if K(x, y) = λ i φ i (x) φ i (y) for an orthonormal basis {φ i } then the trace is the sum of the eigenvalues as expected.

10 Other basic properties of T (a) If K(x, y) 2 dx dy <, then T is a bounded operator on L 2. (b) If the kernel of T 1 is K 1 (x, y) and the kernel of T 2 is K 2 (x, y), then the kernel of T 1 T 2 is given by b a K 1 (x, z) K 2 (z, y) dz. (c) If T has kernel K(x, y), then T has kernel K(y, x).

11 Trace-class operators There is another abstract definition of an operator determinant defined for general operators defined on a general Hilbert space that are of the form I + T where T is a trace class operator, so next we turn to these.

12 We say that T is trace class if T 1 = (TT ) 1/2 e n, e n < n=1 for any set of orthonormal basis vectors {e n } in our Hilbert space. It is not always convenient to check whether or not an operator is trace class from the definition.

13 It is fairly straight forward to check if an operator is Hilbert-Schmidt and it is well known that a product of two Hilbert-Schmidt operators is trace class. The definition of the Hilbert-Schmidt class of operators is the set of S such that the sum Se i, e j 2 < i,j is finite for some choice of orthonormal basis. If it is, then the above sum is independent of the choice of basis and its square root is called the Hilbert-Schmidt norm of the operator, denoted by S 2.

14 Properties of trace class operators (a) Trace class operators form an ideal in the set of all bounded operators and are closed in the topology defined by the trace norm. (b) Hilbert-Schmidt operators form an ideal in the set of all bounded operators with respect to the Hilbert-Schmidt norm and are closed in the topology defined by the Hilbert-Schmidt norm. (c) The product of two Hilbert-Schmidt operators is trace class.

15 (d) If T is trace class, then T is a compact operator. (e) If T is trace class, then λ i < where the λ i s are the eigenvalues of T. (f) Hence the product (1 + λ i ) is always defined and finite and we denote it by det(i + T). (g) If T is trace class, then det P α (I + T)P α det(i + T) for orthogonal projections P α that tend strongly (pointwise) to the identity. (For the first determinant we think of P α (I + T)P α as the operator defined on the image of P α.)

16 (h) If A n A, B n B strongly (pointwise in the Hilbert space) and if T is trace class, then A n TB n ATB in the trace norm. (i) The functions defined by tr T = λ i and det(i + T) are continuous on the set of trace class operators with respect to the trace norm. (j) If T 1 T 2 and T 2 T 1 are trace class then tr (T 1 T 2 ) = tr (T 2 T 1 ) and det(i + T 1 T 2 ) = det(i + T 2 T 1 ).

17 If T is trace class then we may define D(λ) = det(i λt) for all complex λ. So the natural question is: if T is an integral operator with kernel K(x, y), does D(λ) = D(λ)? First we should verify that the integral operator is trace class or even Hilbert-Schmidt. This is not always the case and requires an additional assumption.

18 It is not hard to check that if K(x, y) is continuous on [a, b] [a, b], then T is Hilbert-Schmidt and T 2 = b b a a K(x, y) 2 dxdy. The operator T is trace class if, in addition, the Hilbert-Schmidt norm of the kernel K is finite and then y T 1 T 2 + b a K/ y 2. If the above condition holds then D(λ) = D(λ) since they are both entire functions and agree on a countable set, at all the points 1/λ i corresponding to the eigenvalues of the integral operator.

19 There are other conditions which guarantee that an integral operator is trace class. And whenever that is the case, the same argument can be used to show the definitions of the determinants agree. The next few slides show why this is true for certain convolution operators.

20 The truncated Wiener-Hopf operator W α (σ) is defined by where k is given by f (x) g(x) = f (x) + k(x) = 1 2π α 0 k(x y) f (y) dy, (σ(ξ) 1)e ixξ dξ. The symbol σ is defined on the real line and it is assumed that σ 1 in L 1 (R). If α =, then W α (σ) is the usual Wiener-Hopf operator and will be denoted by W(σ).

21 This last condition assures us that W α (σ) I is trace class. To see this, think of ξ as fixed and e i(x y)ξ (σ(ξ) 1) = e ixξ e iyξ (σ(ξ) 1) as the kernel of a rank one operator. The kernel in question is a L 1 limit of sums of such operators and since trace class operators are closed in the trace norm, the truncated Wiener-Hopf operator is trace class.

22 Properties of W(σ) Suppose σ + 1 H and σ 1 H. (Recall H is the set of all φ in L such that the Fourier transform of φ vanishes on the negative real axis). Then W(σ )W(σ)W(σ + ) = W(σ σσ + ). In particular, if σ is also in H, then W(σ 1 + )W(σ + ) = I. In general for appropriate f, The analogue holds for σ. f (W(σ + )) = W(f (σ + )).

23 The other important property of the operator W(σ) is that W(σ)W(φ) = W(σ φ) H(σ)H( φ) where H(σ) has kernel (σ 1) x+y, H( φ) has kernel (φ 1) x y, and where ψ x denotes the Fourier transform of ψ at x. We also note that H(σ) is Hilbert-Schmidt on L 2 (0, ) if or (σ 1) x+y 2 dx dy <, x (σ 1) x 2 dx <.

24 Our first task will be to compute the Fredholm det W α (σ) exactly, and then from the exact form, compute the asymptotics. We will require that not only σ 1 be in L 1, but that k is as well and that (1 + x ) k(x) 2 dx <. Then if σ is bounded away from zero, has index zero, we can factor σ = σ σ + where σ + 1 and σ 1 are in H, and H, respectively.

25 The conditions on k (or equivalently σ) are closed Banach algebra conditions and thus if σ has a continuous logarithm, then the logarithm also satisfies these conditions, and any H 2 projection of the logarithm. This means we can factor σ = σ σ + and the factors will satisfy the conditions as well. In the next slide Q α is the orthogonal projection of L 2 (0, ) onto L 2 (α, ).

26 Borodin-Okounkov-Geronimo-Case identity This is given by where det(w α (σ)) = G(σ) α E(σ) det(i Q α L(σ) Q α ) G(σ) := 1 2π E(σ) = exp 0 log σ (x) dx, x s(x) s( x)dx and s(x) = (log σ) x. L is a trace class operator acting on L 2 (α, ) with kernel ( ) ( ) σ σ+ L(x, y) = 1 1 dz. 0 σ + x+z σ z y

27 We begin with the observation that W α (σ) = P α W(σ)P α where P α is the orthogonal projection of L 2 (0, ) onto L 2 (0, α), and P α W(σ + ) = P α W(σ + )P α, and W(σ )P α = P α W(σ )P α. Then P α W(σ) P α = P α W(σ + ) W(σ 1 + ) W(σ) W(σ 1 ) W(σ ) P α = P α W(σ + ) P α W(σ 1 + ) W(σ) W(σ 1 ) P α W(σ ) P α. We compute the determinant of each term above.

28 The determinants can be easily seen to be ( α exp 2π using the formula det P α W(σ ± )P α ) log σ ± dx det (exp A) = exp (tr A). Thus det P α W(σ + )P α det P α W(σ )P α ( α ) ( α = exp log σ + + log σ dx = exp 2π 2π ) log σ dx.

29 To address the middle term P α W(σ 1 + ) W(σ) W(σ 1 )P α we denote W(σ 1 + ) W(φ) W(σ 1 ) by A. From our conditions on σ this is an invertible operator. We now use Jacobi s identity det P A P = (det A) (det Q A 1 Q) where P + Q = I and we are left with the computation of two more terms.

30 The determinant det W(σ 1 + ) W(σ) W(σ 1 ) = det W(σ 1 + ) W(σ ) W(σ + )W(σ 1 ) exists by our assumptions on σ. Since it is of the form e B e C e B e C with the commutator of B and C trace class, we can write the determinant as exp(tr [B, C]). Computed it becomes exp 0 x s(x) s( x) dx. (Note: The determinant can also be expressed as det W(σ 1 ) W(σ), an answer that works for matrix-valued symbols as well.)

31 We still need to compute det Q A 1 Q. So A 1 is A = W(σ 1 + ) W(σ )W(σ + ) W(σ 1 ) W(σ ) W(σ 1 + )W(σ 1 ) W(σ + ) = W(σ σ 1 + )W(σ 1 σ + ) and the result follows. Notice the kernel that corresponds to this last operator is ( ) ( ) σ σ+ L(x, y) = 1 1 σ + σ as desired. 0 x+z dz, z y

32 To recap: det P α W(σ ) P α det P α W(σ + ) P α = G(σ) α det A = E(σ) det Q A 1 Q = det(i Q α L(σ) Q α )

33 Since Q α tends strongly to zero, we have the asymptotics (the Kac formula): det(w α (σ)) G(σ) α E(σ) = G(σ) α det W(σ 1 )W(σ). I. Fredholm, Sur une classe d equations functionnelles, Acta Math. 27 (1903) I.C. Gohberg, M.G. Krein, Introduction to the theory of linear nonselfadjoint operators Vol. 18, Translations of Mathematical Monographs, Amer. Math. Soc., Rhode Island, A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin, 1990.

34 A. Borodin, A. Okounkov, Fredholm determinant formula for Toeplitz determinants Int. Eqns. Operator Th. 37 (2000), J. Geronimo, K. Case. Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys. 20 (1979), no. 2, E. Basor, H Widom, On a Toeplitz Identity of Borodin and Okounkov, Int. Eqns. Operator Th 37 (2002), no A. Böttcher, On the determinant formulas by Borodin, Okounkov, Baik, Deift, and Rains, Oper. Th.: Adv. and Appl. 135 (2002), E. Basor, Y. Chen, A note on Wiener-Hopf determinants and the Borodin-Okounkov identity, Int. Eqns. Operator Th., 45 (2003),

35 At times, the B-O-G-C formula can be used to find another term of the expansion. Here is one example without the details. The kernel is g sin π(x y) π sinh g(x y), g > 0. Then as α, det(i Q α L(σ) Q α ) 1 C(g) e 2gα(1 θ/π), where C(g) is a completely determined constant and cos θ = e π2 /g, 0 < θ < π/2.

36 The proof given to find the asymptotics of the determinants of the truncated Wiener-Hopf operators can be easily modified to apply to many other situations. One easy example is to perturb the Wiener-Hopf operators by a trace class integral operator ( on L 2 (0, )). det P α (W(σ) + T)P α G(σ) α det(w(σ 1 )(W(σ) + T)) The other classic example is for finite Toeplitz matrices and many of the ideas for the proof given above were first realized for those matrices. However, there are others, often considered because of problems in random matrix theory.

37 The operator W α (φ) is unitarily equivalent to P α F 1 M φ FP α where F is the Fourier transform and M φ is multiplication by the function φ. For Laguerre ensembles in random matrix theory one might study an analogous operator, the Bessel operator B α (φ) defined on L 2 (0, α) by P α H ν M φ H ν P α where H ν is the Hankel transform of order ν given by H ν (f )(x) = and J ν is the Bessel function of order ν. 0 tx Jν (tx)f (t) dt,

38 In more familiar kernel form we have the operator with kernel φ(y) J ν( x) yj ν ( y) xj ν ( x)j ν ( y). 2(x y) Using much the same ideas for the proof as we did for the Wiener-Hopf case, one can show that if φ = e b 1, ( α det(i+b α (φ)) exp b(x)dx ν 2π ) x(ˆb(x)) 2 dx. 2 0

39 If we scale in Gaussian Unitary ensembles on the edge of the spectrum, linear statistics problems reduce to the study of the Airy operators, integral operators on L 2 (0, α) with kernel A α (f )(x, y) = f (x/α) where A(x) is the Airy function. 0 A(x + z)a(z + y)dz (This operator also has an equivalent definition in terms of multiplication and an Airy transform.)

40 The asymptotic formula reads det(i + A α (f )) exp ( c 1 α 3/2 + c 2 ) where and c 1 = 1 π 0 x log(1 + f ( x)) dx c 2 = 1 x G(x) 2 dx 2 0 G(x) = 1 e ixy log(1 + f ( y 2 ))dy. 2π

41 All that we have done so far requires a nicely behaved symbol σ. So what happens for more general σ? For example, the familiar sine kernel sin π(x y) π(x y) has inverse Fourier transform that is a characteristic function of [ 1, 1], and thus does not satisfy the conditions that are assumed for the Kac formula. To be able to handle this case, in at least some situtations, we turn to Toeplitz determinants.

42 The analogue of what we have just done in the Toeplitz case is the strong Szegö limit theorem. It states that if the symbol ϕ defined on the unit circle has a sufficiently well-behaved logarithm then the determinant of the Toeplitz matrix T n (ϕ) = (ϕ j k ) j,k=0,,n 1 has the asymptotic behavior D n (φ) = det T n (ϕ) G(ϕ) n E(ϕ) as n where ( ) G(ϕ) = e (log ϕ) 0, E(ϕ) = exp k (log ϕ) k (log ϕ) k. k=1 Here subscripts denote Fourier coefficients.

43 In 1968, Fisher and Hartwig raised a conjecture about a certain class of Toeplitz matrices with singular symbols. If ϕ γ,β (e iθ ) = (2 2 cos θ) (γ+β)/2 e i(θ π)(γ β)/2 for 0 < θ < 2π (this symbol is said to have a pure Fisher-Hartwig singularity), then their symbols had the form ψ(z) = ϕ(z) N ϕ γj,β j (z/z j ) where ϕ satisfies the assumption of Szegö s theorem and z 1,, z N are distinct points on the unit circle. j=1

44 They conjectured that for some range of the parameters the asymptotics had the form det T n (ψ) G(ψ) n n γ j β j E(ϕ, γ j, β j, z j ) where E(ϕ, γ j, β j, z j ) is a constant (whose value they did not conjecture). The conjecture has a long history and due to the work of many mathematicians the constant E(ϕ, γ j, β j, z j ) was determined.

45 The conjecture has now been proved in great generality provided that Re (γ j ± β j ) < 1. The basic technique is to first show that the quotient D n (φψ)/d n (φ)d n (ψ) tends to a limit if the singularities of the symbols do not overlap (a localization theorem) and then to compute the determinant exactly for the pure singularity which fortunately can be done. ϕ γ,β

46 It is natural to ask what happens to non-smooth symbols in the Wiener-Hopf case as well. We take as the analogue of the pure Fisher-Hartwig singularity the symbol σ γ,β (ξ) = ( ξ 0i ξ i ) γ ( ) β ξ + 0i. ξ + i (We specify the arguments of ξ ± 0i and ξ ± i to be zero or close to it when ξ is large and positive.) This has the behavior σ γ,β (ξ) ξ γ+β e 1 2 iπ(γ β) sgnξ as ξ 0.

47 The general symbol is then of the form σ 0 (ξ) r σ γr,β r (ξ ξ r ) where σ 0 is a symbol for which the Strong Szegö theorem holds. For an attempted proof, it seems reasonable to try to do what was done in the Toeplitz case, that is, (1) devise the proper localization techniques (2) then try to evaluate the determinants exactly in the case of the pure singularity.

48 While the first step has been accomplished, the second step, except in some special cases has never been done. So something else is needed to overcome this obstacle. Now it turns out the key is to use an identity for our smooth symbols, the Borodin-Okounkov-Geronimo-Case Identity. Note that for Toeplitz determinants the analogue of the identity follows just as before.

49 Before we show how these identities link the Toeplitz and Wiener-Hopf determinants there is an extra complication since for Wiener-Hopf symbols of the form σ γ,β the Wiener-Hopf operator is not of the from I + T where T is trace class, but rather T is Hilbert-Schmidt. Thus we need a regularized determinant version of these identities. This determinant for operators of the form I + T where T is Hilbert-Schmidt with eigenvlaues λ i is defined by: (1 + λi )e λ i.

50 This is quite easy to do and the corresponding statements are: For the regularized determinant we have det 2 T n (ϕ) = G 2 (ϕ) n E(ϕ) det (I K n ) where G 2 (ϕ) = exp ( (log ϕ) 0 ϕ ). For Toeplitz determinants this follows immediately since holds for any finite matrix A. det 2 A = det A e tr(a I)

51 The Wiener-Hopf analogue is where det 2 W α (σ) = G 2 (σ) α E(σ) det (I Q α L Q α ) ( 1 G 2 (σ) = exp 2π ) (log σ(ζ) σ(ζ) + 1) dζ.

52 We can state the main result: If Re (γ ± β) < 1 then det 2 W α (σ γ,β )/G 2 (σ γ,β ) α det 2 T n (ϕ γ,β )/G 2 (ϕ γ,β ) n when α 2n.

53 The asymptotics of det 2 T n (ϕ γ,β )/G 2 (ϕ γ,β ) n are well-known and given by the formula γβ G(1 + γ)g(1 + β) n G(1 + γ + β) where G(1 + z) is the Barnes G-function. G(1 + z) is an entire function satisfying G(1 + z) = Γ(z)G(z) and defined by the infinite product e (1+γ)λ2 n=1 where γ is Euler s constant. (1 + λ2 n 2 )n e λ2 /n

54 Thus in the Wiener-Hopf case det 2 W α (σ γ,β ) G 2 (σ γ,β ) α γβ G(1 + γ)g(1 + β) (α/2). G(1 + γ + β) When β = γ the Wiener-Hopf operators have a well-defined determinant and thus we have as a corollary: If Re (γ) < 1/2 then when α 2n. det W α (σ γ,γ ) e αγ det T n (ϕ γ,γ )

55 Here is the idea of the proof: The function ϕ γ,β = (1 z) γ (1 1 z )β z = e iθ. Define Then ϕ r,γ,β = (1 rz) γ (1 r z )β z = e iθ. D n (ϕ γ,β ) = lim r 1 D n (ϕ r,γ,β ). Apply the identity to the term D n (ϕ r,γ,β ) and take the limit as r 1. Do the same for W α (σ γ,β ) and then compare the answers.

56 To get a complete answer one needs to paste together the asymptotics of the pieces and the quotient terms. This has been done in some cases, but not all. For certain piecewise continuous symbols here is the complete answer. Suppose that σ is bounded away from zero, piecewise C 2 with a finite number of jump discontinuities at the points x 1,..., x R, has a appropriately defined argument which vanishes at ± and assume that σ L 1, (1 + x 2 )σ (x) L 2.

57 Then det(w α (σ)) G(σ) α (α) r λ2 r E(σ) g(λ r ), r where G(σ) = exp 1 log φ(x) dx 2π λ r = 1 2π log(σ(x r+)/σ(x r )), g(λ) = G(1 + λ)g(1 λ) 1 e x E(σ) = exp {x log σ x log σ x λ 2 x r} dx. 0 r

58 M. E. Fisher, R. E. Hartwig, Toeplitz determinants: some applications, theorems, and conjecture, Adv. Chem. Phys. 15 (1968) T. Ehrhardt, A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities, Oper. Theory Adv. Appl. 124, (2001), E. Basor, H. Widom, Wiener-Hopf determinants with Fisher-Hartwig symbols, Operator Th: Advances and App. 147 (2004) P. Deift, A. Its, I. Krasovsky, Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. of Math (2) 174 (2011), no. 2,

59 Some applications to random matrix theory In Random Matrix Theory (RMT) there are three important invariant ensembles. 1. Unitary Ensembles (UE) which are n n Hermitian matrices together with a distribution that is invariant under unitary conjugation. In other words, if U is any unitary matrix, the measure of a set S of matrices is the same as US U. 2. Orthogonal Ensembles (OE) which are n n symmetric matrices together with a distribution that is invariant under conjugation by an orthgonal matrix.

60 3. Symplectic ensembles (SE) which are 2n 2n Hermitian self-dual matrices together with a distribution that is invariant under unitary-symplectic conjugation, that is, matrices M that satisfy M = M = JM t J t and distributions invariant under the mapping M UMU with U satisfying UU = I and UJU = J where J = diag(j 2,..., J 2 ), J 2 = ( ).

61 Each has an associated probability distribution of the form P n (M) dm = 1 Z n e tr Q(M) dm where dm is Lebesgue measure on the algebraically independent entries of M, Q is a real-valued function, generally an even degree polynomial, and Z n is a normalization constant. In the case that Q(x) = x 2 these measures are equivalent to having, as much as is algebraically possible, matrices whose entries are independent normal or Gaussian random variables.

62 These ensembles are probably the most studied and are indicated with an extra Gaussian adjective as the Gaussian Unitary Ensemble (GUE), the Gaussian Orthogonal Ensemble (GOE), and the Gaussian Symplectic Ensemble (GSE). In each of the above cases one can compute the induced distribution on the space of eigenvalues. For example, in the GUE case one can diagonalize a Hermitian matrix as UDU, where U is unitary and D is a diagonal matrix and then make a change of variables M (U, D). After integrating out the unitary part (which is equivalent to computing a Jacobian) one arrives at an induced distribution on the space of eigenvalues.

63 A similar computation can be made for all three ensembles and the resulting probability densities have the form c n e β n 2 i=1 Q(xi) (x i ) β where (x i ) is the Vandermonde determinant (x i ) = det(x i 1 j ) 1 i,j n = j<i (x i x j ) and c n is the normalizing constant.

64 Here, more precisely, we mean that if f is any symmetric function of n real variables, then the expected value of f is c n f (x 1, x 2,..., x n )e β n 2 i=1 Q(xi) (x i ) β dx 1 dx 2... dx n. R n For UE β = 2, for OE β = 1, and for SE β = 4, and thus the three ensembles are often referred to as the β = 1, 2, or 4 ensembles. (It can be shown for SE that eigenvalues occur in pairs and thus all three densities are defined as functions of n variables.)

65 As soon as the densities are known, one can try to compute some statistical information about the eigenvalues. Since it is easier to describe, we illustrate the ideas with GUE. We can first factor the exponential terms into the Vandermonde determinant so that the entry xj i 1 is replaced by e x2 j /2 xj i 1 and then using elementary row operations we can replace each row by any polynomial. That is, replace e x2 j /2 xj i 1 by e x2 j /2 p i 1 (x j ) only changing the determinant by a constant factor.

66 So we choose to replace them by the normalized Hermite polynomials h k (x) which satisfy h k (x) h j (x) e x2 dx = δ jk. From this it follows, after identifying the constant, that the density on the space of eigenvalues for GUE is c n e n i=1 x2 i (xi ) 2 = 1 n! det K n(x i, x j ) where n 1 K n (x i, x j ) = ϕ k (x i )ϕ k (x j ), k=0 and ϕ k (x) = h k (x)e x2 2.

67 The Christoffel-Darboux formula allows one to analyze K n (x, y) in a more concise form since it says that for x y, n φ n (x)φ n 1 (y) φ n (y)φ n 1 (x) K n (x, y) =, 2 x y and for x = y, K n (x, y) is the limit of this expression as x y. All the information we need is somehow contained in the function K n (x, y). It is clear that for large n this information is intimately related to knowledge about the asymptotics of the Hermite polynomials.

68 For example, the density of eigenvalues, ρ n (x), defined to be the limit of the expected number of eigenvalues in an interval around x divided by its length, as the length tends to zero, is exactly K n (x, x). Using the known asymptotics for the Hermite polynomials one can show that lim n 2 n ρ n( { 2 2n x ) = π 1 x 2 if x < 1 0 if x > 1 holds uniformly on compact sets of x < 1 and x > 1. This result, one of the first successes of Wigner s program is called the Wigner semi-circle law.

69 The semi-circle law tells us how we should rescale so that we can obtain meaningful answers. Thus we replace K n (x, y) with ( ) 1 x y K n,. 2n 2n 2n From the theory of Hermite polynomials one can show that as n, ( ) 1 x y sin(x y) K n, 2n 2n 2n π(x y).

70 Now consider a random variable of the form n f (x i 2n). i=1 This is called a linear statistic. A fundamental formula from probability theory shows that if we call the probability distribution function φ n, then its inverse Fourier transform is given by e ik n j=1 f (x j 2n) K n (x 1,..., x n ) dx 1 dx n.

71 The transform is then 1 + (e ik n j=1 f (x j 2n) 1) K n (x 1,..., x n ) dx 1 dx n. We expand the exponential and collect terms in k using the helpful property that n! K n (x 1,..., x m, x m+1,..., x n ) dx m+1 dx m (n m)! = det K n (x i, x j ) m i,j=1.

72 The expansion immediately shows that this is the Fredholm determinant for the kernel (e ik n j=1 f (x j 2n) 1) K n (x, y). If we change variables let n then the limit is the Fredholm determinant det(i + T) where T has kernel K(x, y) = (e i k f (x) sin(x y) 1) π(x y). This has the same Fredholm determinant as the operator with K(x, y) = sin(x y) π(x y) (ei k f (y) 1).

73 We recall that sin x/πx is the Fourier transform of the characteristic function of the interval ( 1, 1). Thus we can write our kernel as K(x, y) = 1 2π χ(ξ)e i(x y)ξ (e i k f (y) 1) dξ where χ(ξ) is the characteristic function of the interval. But this corresponds to the operator that sends g(y) to g 1 1 e iξx e iyξ (e i k f (y) 1)g(y) dy dξ. 2π 1

74 However, g 1 1 e iξx e iyξ (e i k f (y) 1)g(y) dy dξ 2π 1 is the same as FPF 1 M φ and thus has the same determinant as PF 1 M φ FP and is thus a Wiener-Hopf operator with symbol e i k f (y) and restricted to the interval ( 1, 1).

75 Now we take f (x) to be of the form f (x/α) with α a parameter. This is equivalent to changing the interval to ( α, α). Using our limit theorem we have that { α F(φ)(k) exp i k f (x) dx k 2 π 0 } x F 1 (f (x)) 2 dx. This tells us that asymptotically the distribution function is Gaussian and identifies the mean and variance.

76 But what happens if f is not smooth? Then we need to use Fisher - Hartwig symbols. Suppose we let f be the function that counts the number of eigenvalues in a large interval, f (x/α) = χ (α,α) (x). This yields a piecewise continuous symbol for e i k f (x).

77 To begin, let us compute the mean of the linear statistic before scaling. µ = f (xi 2n/α)Kn (x 1,..., x n ) dx = n = f (x 1 2n/α)Kn (x i,..., x n ) dx f (x 1 2n/α)Kn (x 1, x 1 ) dx. Changing variables in the limit we have µ = 2α π.

78 We can do a similar computation and find that the variance is asymptotically or 1 π log 2 α + 1 (1 + γ + log 2) 2 π2 1 log 2 α + O(1) π2

79 Recall our formula where det(w α (σ)) G(σ) α (α) r λ2 r E(σ) g(λ r ), G(σ) = exp 1 log φ(x) dx 2π λ r = 1 2π log(σ(x r+)/σ(x r )) g(λ) = G(1 + λ)g(1 λ) E(σ) = exp 0 {x log σ x log σ x 1 e x x r λ 2 r} dx. r

80 The symbol of interest is 1 if ξ > 1, σ(ξ) = e ik if ξ < 1, G(σ) = exp 1 π log φ(x) dx = exp 2 ki π. We will have two jumps with parameters ik 2π replaced by 2α. and ik 2π and α is In this case R = 2 and λ 1 = i k 2π and λ 2 = i k 2π. Thus, (2α) r λ2 r = exp( k2 log 2 α). 2π2

81 Notice that both of these terms do have the property that their logarithms are quadratic in k. This holds true for E(σ) as well. This term is given by exp{ k2 sin 2 x π 2 0 x 1 e x dz} = exp( k2 log 2 ). 2x 2π 2

82 Thus the entire expansion is given by the formula exp{ 2kiα π ( k2 k2 ) log α ( 2π2 π ) log 2}g(i 2 k/2π)2. The presence of this last term involving the function g clearly shows that this expansion does not have a logarithm quadratic in k. Thus, at first it seems the Gaussian nature of the distribution is missing. However, the natural rescaling of the variable: π ( f (x i µ) log 2α yields exp( k 2 /2) as expected.

83 These techniques can be applied to other ensembles as well. We mention only one here. One can follow a similar path to find analogous asymptotic formulas for the truncated Wiener-Hopf + Hankel operators W α (σ) + H α (σ) in a case where σ is the characteristic function of an interval and so has two jump discontinuities.

84 The corresponding kernel is ( sin (x y) (e 2πik 1) π (x y) With σ as above we have, for Re k < 1/2, ) sin (x + y) +. π (x + y) det (W α (σ) + H α (σ)) e 2iαk α 3k2 2 4 k2 G(1 2k) G(1 + 2k), where G is the Barnes G-function.

85 Gap probabilites Another important statistic in RMT is the gap probability, which is the probability that no eigenvalues are in the interval (a, b). Using algebraic properties of K n as before one can show that in the limit the gap probability is given by a Fredholm determinant on L 2 (a, b) of the form det(i λt) with kernel K(x, y) = sin(x y) π(x y).

86 If we are interested in applying the limit theorems as before, we must restrict ourselves to small λ since the symbol will vanish on an interval. In a wonderful connection to differential equations, it is known by the work of Jimbo, Miwa, Môri, and Sato that if we define σ(s) = d log det(i K) ds then σ satisfies a second-order nonlinear differential equation of Painlevé type. The theory of Painlevé equations and the theory of integrable systems then yield information about the asymptotics of the probability distribution. Our goal now is to describe the operator approach to the equations.

87 In the following slides we consider det(i λk) where K is the sine kernel and we think of the operator as defined on L 2 ( s, s). In other words, we want information about the probability of finding no eigenvalues in the interval ( s, s). It is useful not to try to compute the determinant directly, but the log of the determinant. This is because we have the formula log det(i λk) = trace log(i λk) at our disposal. We will eventually derive a differential equation (nonlinear) that has a connection to the above function (thought of as a function of s).

88 Suppose {K (s)} is a family of operators, such that K (s) is trace class and dk(s) ds is defined. Then whenever this makes sense, d ( ) ds (log (det (I K (s)))) = trace (I K (s)) 1 K (s). This is easy to verify since the trace is linear and tr AB = tr BA.

89 Now let J be the interval ( s, s) and consider the operator K (s) with kernel K (x, y) χ J (y) where K (x, y) is any continuous kernel. This operator sends f to the function s s K (x, y) f (y) dy. To find K (s) we use the Fundamental Theorem of Calculus.

90 The operator K (s) sends f K (x, s) f (s) + K (x, s) f ( s). Note that the image of f is finite rank and spanned by the two vectors, K (x, s) and K (x, s). In terms of kernels we have K (x, s) δ (y s) + K (x, s) δ (y + s). This operator does not make sense for all functions in L 2 but only for functions that are continuous. We will ignore this fact for the time being.

91 We now define R(x, y) to be the resolvent kernel, that is, the kernel that corresponds to the operator Since (I K) 1 K. (I K) 1 = I + (I K) 1 K the operator (I K) 1 has kernel ρ (x, y) = δ (x y) + R (x, y).

92 The kernel of (I K) 1 K (s) is given by ρ(x, z){k (z, s) δ (y s) + K (z, s) δ (y + s)} dz which is the same as R (x, s) δ (y s) + R (x, s) δ (y + s). This has rank two, just like the image of K (s). We can also easily compute its trace to see that it is R (s, s) + R ( s, s).

93 To summarize the list of kernels so far: operator kernel K (s) K (x, s) δ (y s) + K (x, s) δ (y + s) (I K) 1 ρ (x, y) = δ (x y) + R (x, y) (I K) 1 K (s) R (x, s) δ (y s) + R (x, s) δ (y + s)

94 We now compute the kernels of some additional operators. The operator d ds (I K) 1 has kernel R (x, s) ρ (s, y) + R (x, s) ρ ( s, y) = M (x, y). This is because d ds (I K) 1 = (I K) 1 dk ds (I K) 1. The right hand side has kernel ρ (x, y) {K (y, s) δ (z s)+k (y, s) δ (z + s)}ρ (z, u) dy dz or as desired. R (x, s) ρ (s, u) + R (x, s) ρ ( s, u)

95 Up to this point we have used no information about the sine kernel in particular, but from now on, we will use the properties of the sine kernel. The commutator of two operators A and B, AB BA and is denoted by [A, B], and we let D be the derivative operator. The operator [D, (I K) 1 ] has kernel R (x, s) ρ (s, y) + R (x, s) ρ ( s, y). To see this, the commutator [ D, (I K) 1] = (I K) 1 [D, K] (I K) 1. DK has kernel K(x,y), but KD is more complicated. x

96 The operator KD operates on a function f by so K D has kernel K D f = s s = (K (x, y) f (y) y=s y= s s K (x, y) f (y) dy s K (x, y) f (y) dy y K (x, s) δ (y s) K (x, s) δ (y + s) Since K (x, y) is a function of x y, K = K x y kernel of [D, K] is K (x, y). y K (x, s) δ (y s) + K (x, s) δ (y + s). and thus the

97 To find the kernel of [D, (I K) 1 ] we have ρ (x, y) [ K (y, s) δ (z s) + K (y, s) δ (z + s)] ρ (z, u) dy dz = ( R (x, s) δ (z s) ρ (z, u) + R (x, s) δ (z + s) ρ (z, u)) dz = R (x, s) ρ (s, u) + R (x, s) ρ ( s, u).

98 To summarize again the list of kernels: operator kernel K (s) K (x, s) δ (y s) + K (x, s) δ (y + s) (I K) 1 ρ (x, y) = δ (x y) + R (x, y) (I K) 1 K (s) R (x, s) δ (y s) + R (x, s) δ (y + s) d (I ds K) 1 R (x, s) ρ (s, y) + R (x, s) ρ ( s, y) [D, (I K) 1 ] R (x, s) ρ (s, y) + R (x, s) ρ ( s, y)

99 We will consider a slightly more general interval in what follows since it is not any harder to do so. We let I be the unions of the intervals (a 2i+1, a 2i ). We write our kernel λk (x, y) = where A (x) = λ sin (x y) π (x y) = A (x) A (y) A (y) A (x) x y λ sin x. We define functions π ( ) Q (x, â) = (I K) 1 A (x) ) P (x, â) = ((I K) 1 A (x) where â is a vector containing each a i. Define the operator M x (Multiplication by x) to be M x f = x f.

100 More kernels! The operator [ M x, (I K) 1] has kernel ) ( ) Q (x, â) ((I K t ) 1 A (y) P (x, â) (I K t ) 1 A (y) where K t has kernel K (y, x). To see this, once again, [ M x, (I K) 1] = (I K) 1 [M x, K] (I K) 1.

101 Now [M x, K] has kernel (A (x) A (y) A (x) A (y)). One half of the kernel of I I I [ M x, (I K) 1] is given by ρ (w, x) A (x) A (y) ρ (y, z) dx dy = ρ (w, x) A (x) dx A (y) ρ (y, z) dy = Q (x, â) I ( ) (I K t ) 1 A (z)

102 while similarly, for the other half ρ (w, x) A (y) A (x) ρ (y, z) dx dy I I ( ) = P (x, â) (I K t ) 1 A (z). And thus the kernel of [ M x, (I K) 1] is the sum of these ) ( ) Q (x, â) ((I K t ) 1 A (z) P (x, â) (I K t ) 1 A (z).

103 The function R (x, y) can now be written in terms of P and Q and is given by: ) ( ) Q (x, â) ((I K t ) 1 A (y) P (x, â) (I K t ) 1 A (y). x y But clearly for our sine kernel the transpose K t is the same as K. So we have thus proved that for x and y both in I, R (x, y) = Q (x, â) P (y, â) P (x, â) Q (y, â). x y

104 Notice the resolvent kernel has the same form as the sine kernel and now we have reduced our problem into finding information about P and Q. In order to determine R (x, x) consider P (y, â) as y approaches a fixed value for x. We can expand P (y, â) in a Taylor series about the point x as ( ) d [ P (y, â) = P (x, â) + P (x, â) (y x) + O (y x) 2]. dx Similarly for Q Q (y, â) = Q (x, â) + ( ) d [ Q (x, â) (y x) + O (y x) 2]. dx

105 Substituting this into the expression for R (x, y), R (x, y) = Q (x, â) P (x, â) + P (x, â) Q (x, â) + O (x y). Taking the limit as y x, R (x, x) = Q (x, â) P (x, â) + P (x, â) Q (x, â). Now let s examine how R behaves at the endpoints of the intervals in I.

106 We define q j as Q(x, â) x=aj and p j as P(x, â) x=aj. For a i a j, R (a j, a k ) = q jp k p j q k a j a k. We can generalize our previous computation to see that [D, (I K) 1] has kernel 2m k=1 ( 1) k R (x, a k ) ρ (a k, y).

107 Using our kernel representations, it follows that Q (x, a) x=aj = p j 2m k=1 ( 1) k R (a j, a k ) q k. Similarly for P (x, â), P (x, a) x=aj = q j 2m k=1 ( 1) k R (a j, a k ) p k.

108 Placing both of these expressions into the formula for R (x, x) yields ) 2m R (a j, a j ) = p j (p j ( 1) k R (a j, a k ) q k = p 2 j + q 2 j = p 2 j + q 2 j k=1 q j ( q j 2m k=1 2m k=1 2m k=1 ( 1) k R (a j, a k ) p k ) ( 1) k R (a j, a k ) (p j q k q j p k ) ( 1) k R (a j, a k ) R (a k, a j ) (a k a j ).

109 The expression for q j a j can be found to be q j a j = p j 2m k = 1 k j ( 1) k R (a j, a k ) q k. In a completely similar fashion, p j a j = q j 2m k = 1 k j ( 1) k R (a j, a k ) p k.

110 Now let s specialize to the case where K (x, y) = sin(x y) and π(x y) where I is ( s, s). For this interval, K is symmetric with respect to interchange of variables and satisfies K (x, y) = K ( x, y). This same property holds for the kernel ρ. To see this define a flip operator J that sends the function f (x) to g(x) = g( x). It is easy to check that if K and J commute. Now (I K) 1 J = J (I K) 1 since (I K) 1 J(I K) = J (I K) 1 (I K).

111 This also tells us that R(x, y) = R( x, y) and thus R(s, s) = R( s, s). It is also the case that : q 1 = q 2. q 1 = lim Q (x, â) = lim + x s s = lim x s s s = lim x s s s = lim x s = lim x s s s x s + s s ρ ( x, y) A (y) dy ρ (x, y) A (y) dy ρ (x, y) A ( y) dy s A similar argument leads to p 1 = p 2. ρ (x, y) A (y) dy = q 2. ρ (x, y) A (y) dy

112 We are finally very close to finding our differential equation. Define a (s) = sr (s, s) b (s) = sr ( s, s). First notice that d ds (sr ( s, s)) = d ds (p 1q 1 ) = p 1 (p 1 2R ( s, s) q 1 ) + ( q 1 + 2R ( s, s) p 1 ) q 1 = p 2 1 q 2 1.

113 Using the same sort of computation one can also show that d ds (sr (s, s)) = p2 1 + q 2 1. If we square both sides of the right-hand side of the last two equations, it is clear that ( ) 2 da = ds ( ) 2 db + 4b 2. ds

114 It is also the case from the fundamental formulas for a and b that a s da ds = 2b2. Differentiating this last equation we see that sa = 4bb. Inserting this into the previous equation we arrive at the following.

115 Theorem The function a satisfies the following differential equation: s 2 (a ) 2 = 8(sa a)( 2(sa a) + (a ) 2 ). We know that if λ is small, then the asymptotics are of the form: log det(i λk) = c 1 s c 2 log s + c 3 + o(1) from our previous results.

116 The asymptotic formula is given here for λ = 1 reads: log det(i K) = s2 2 log s 4 + log ζ ( 1) + o(1). This was first conjectured by Freeman Dyson in The first two terms were proved in 1995 by Harold Widom and the constant term was computed independently by Torsten Ehrhardt in 2007 and also by Deift, Its, Krasovsky, and Zhou. Ehrhardt s techniques involved operator theory methods while the other approach used the Riemann Hilbert approach.

117 E. W. Barnes. The theory of the G-function, Quart. J. Pure and Appl. Math. 31 (1900) E. L. Basor. Distribution Functions for Random Variables for Ensembles of Positive Hermitian Matrices, Comm. Math. Phys. 188 (1997), E. L. Basor, T. Ehrhardt, H. Widom. On the determinant of a certain Wiener-Hopf + Hankel operator, Integral Equations and Operator Theory 47 (2003) E. L. Basor, H Widom. Determinants of Airy operators and applications to random matrices. J. Statist. Phys. 96 (1999) no. 1-2, A. Böttcher, B. Silbermann. Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag, Berlin, 1998.

118 T. Ehrhardt, Dyson s constants in the asymptotics of the determinants of Wiener-Hopf-Hankel operators with the sine kernel, Comm. Math. Phys. 272 (2007), no. 3, 683Ð698. P. Deift, A. Its, I. Krasovsky, X. Zhou, The Widom-Dyson constant for the gap probability in random matrix theory., J. Comput. Appl. Math. 202 (2007), no. 1, 26Ð47. C. Hughes, J. P. Keating, N. O Connell. On the characteristic polynomial of a random unitary matrix, Commun. Math. Phys. 220 (2001) M. Jimbo, T. Miwa, Y. Môri, M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica D 1 (1980), M. Kac. Toeplitz matrices, translation kernels, and a related problem in probability theory, Duke Math. J. 21 (1954),

119 M. L. Mehta. Random Matrices, Academic Press, Rev. and enlarged 2nd ed., San Diego, C. A. Tracy, H. Widom. Level-Spacing Distributions and the Airy Kernel, Comm. Math. Phys. 159 (1994), C. A. Tracy, H. Widom. Introduction to random matrices, in Proc. 8th Scheveningen Conf., Springer Lecture Notes in Physics, C. A. Tracy, H.Widom. Level spacing distributions and the Bessel kernel, Commun. Math Phys. 161 (1994) H. Widom. Asymptotic behavior of block Toeplitz matrices and determinants. II, Adv. in Math. 21:1 (1976) 1-29.