Fredholm determinant with the confluent hypergeometric kernel

Transcription

1 Fredholm determinant with the confluent hypergeometric kernel J. Vasylevska joint work with I. Krasovsky Brunel University Dec 19, 2008

2 Problem Statement Let K s be the integral operator acting on L 2 ( s, s) with confluent hypergeometric kernel: K s (u, v) = 1 2π Γ(1 β + α)γ(1 + β + α) Γ(1 + 2α)Γ(2 + 2α) A(u)B(v) A(v)B(u), u v A(x) = 2e πiβsgn(x)/2 x 2x α e ix φ(α + β + 1, 2α + 2, 2ix), B(x) = e πiβsgn(x)/2 2x α e ix φ(α + β, 2α, 2ix), φ(a, c, z) is the confluent hypergeometric function; α, β are complex parameters, Rα > 1/2. Our aim is to obtain the asymptotics of the Fredholm determinant P s = det(i K s ) for large s.

3 Motivation The kernel K s is a particular case of a kernel that appears in the representation theory of the infinite-dimensional unitary group considered by Borodin and Olshanski. K s can be obtained as a scaling limit at a Fisher-Hartwig singularity for a unitary random matrix ensemble. (i.e. a root-type singularity and a step-function) if β = 0 K s reduces to the kernel in terms of Bessel functions: uv J α+ 1 (u)j 2 α 1 (v) J 2 α+ 1 (v)j 2 α 1 (u) 2 K s (u, v) =, 2 u v that was considered by Nagao, Slevin, Akemann et all, Witte, Forrester, Kuijlaars, Vanlessen. It can be obtained as a scaling limit at a root-type singular point for a unitary random matrix ensemble. if α = 0, β = 0 K s becomes the sine-kernel: K sin (x, y) = sin(x y) π(x y)

4 Theorem1 Theorem (1) The asymptotics for the Fredholm determinant P s = det(i K s ) on ( s, s) as s are given by the formula ( ) ln P s = s αs 4 + α2 β 2 ln s 2α 2 ln 2 ln G(1 + α + β)g(1 + α β) G(1 + 2α) + c 0 + O ( ) 1, where s c 0 = 1 12 ln 2 + 3ζ ( 1) = 2 ln G(1/2) + 1 ln π. 2

5 Theorem1 Theorem (1) The asymptotics for the Fredholm determinant P s = det(i K s ) on ( s, s) as s are given by the formula ( ) ln P s = s αs 4 + α2 β 2 ln s 2α 2 ln 2 ln G(1 + α + β)g(1 + α β) G(1 + 2α) + c 0 + O ( ) 1, where s c 0 = 1 12 ln 2 + 3ζ ( 1) = 2 ln G(1/2) + 1 ln π. 2 comparison with the sine-kernel asymptotics: ln det(i K sin ) = s2 2 1 ( ) 1 4 ln s + c 0 + O, s, s (des Cloizeaux and Mehta (1973), Dyson (1976), Widom (1994), Deift, Its, Zhou (1997), Krasovsky (2004), Ehrhardt(2007))

6 Two usual types of endpoints the density of eigenvalues vanishes as a square root ( soft edge of the spectrum, e.g., GUE endpoints of semicircle). In the scaling limit at the endpoint one obtains the Airy kernel: K Airy (x, y) = Ai (x)ai (y) Ai (y)ai (x), on ( s, ) x y

7 Two usual types of endpoints the density of eigenvalues vanishes as a square root ( soft edge of the spectrum, e.g., GUE endpoints of semicircle). In the scaling limit at the endpoint one obtains the Airy kernel: K Airy (x, y) = Ai (x)ai (y) Ai (y)ai (x), on ( s, ) x y Asymptotics of the Tracy-Widom distribution: ln det(i K Airy ) = s ln s + κ + O(s 3/2 ), s +, κ = 1 24 ln 2 + ζ ( 1), (Tracy, Widom (1994), Deift, Its, Krasovsky (2008), Baik, Buckingham, DiFranco (2008))

8 Theorem2 the density of eigenvalues diverges as a square root ( hard edge of the spectrum, e.g. the Laguerre ensemble at 0 or Jacobi ensemble at the edgepoints). In the scaling limit at the endpoint one obtains the Bessel kernel: xja+1 ( x)j a ( y) yj a ( x)j a+1 ( y) K Bes (x, y) =, (a > 1), 2(x y) on (0, s). (Tracy, Widom (1994))

9 Theorem2 the density of eigenvalues diverges as a square root ( hard edge of the spectrum, e.g. the Laguerre ensemble at 0 or Jacobi ensemble at the edgepoints). In the scaling limit at the endpoint one obtains the Bessel kernel: xja+1 ( x)j a ( y) yj a ( x)j a+1 ( y) K Bes (x, y) =, (a > 1), 2(x y) on (0, s). (Tracy, Widom (1994)) Theorem (2) The large s asymptotics of P s = det(i K Bes ) are given by ln det(i K Bes ) = s ( ) 4 + a s a2 1 4 ln s + c 1 + O, s where c 1 = ln G(1 + a) (2π) a/2.

10 sketch of the proof for the Th.1 The main idea is to use a double-scaling limit of a Toeplitz determinant to obtain asymptotics of the Fredholm determinant P s. The Toeplitz determinant with symbol f is given by the expression: ( 1 2π ) n 1 D n (f) = det e i(j k)θ f(θ)dθ 2π 0. j,k=0

11 sketch of the proof for the Th.1 The main idea is to use a double-scaling limit of a Toeplitz determinant to obtain asymptotics of the Fredholm determinant P s. The Toeplitz determinant with symbol f is given by the expression: ( 1 2π ) n 1 D n (f) = det e i(j k)θ f(θ)dθ 2π Let f ϕ (θ) = 0. j,k=0 { f(z) = z 1 2α z β e iπβ, z = e iθ, ϕ < θ < 2π ϕ, 0, otherwise,

12 sketch of the proof for the Th.1 The main idea is to use a double-scaling limit of a Toeplitz determinant to obtain asymptotics of the Fredholm determinant P s. The Toeplitz determinant with symbol f is given by the expression: ( 1 2π ) n 1 D n (f) = det e i(j k)θ f(θ)dθ 2π Let f ϕ (θ) = 0. j,k=0 { f(z) = z 1 2α z β e iπβ, z = e iθ, ϕ < θ < 2π ϕ, 0, otherwise, Then D n (f 2s n P s = lim n D n (f) ).

13 sketch of the proof for the Th.1 The main idea is to use a double-scaling limit of a Toeplitz determinant to obtain asymptotics of the Fredholm determinant P s. The Toeplitz determinant with symbol f is given by the expression: ( 1 2π ) n 1 D n (f) = det e i(j k)θ f(θ)dθ 2π Let f ϕ (θ) = 0. j,k=0 { f(z) = z 1 2α z β e iπβ, z = e iθ, ϕ < θ < 2π ϕ, 0, otherwise, Then D n (f 2s n P s = lim n D n (f) ). Here the denominator is known (the Toeplitz determinant with a Fisher- Hartwig singularity, Böttcher, Silbermann): D n (f) = n α2 2 β G(1 + α + β)g(1 + α β) (1 + o(1)), n. G(1 + 2α)

14 sketch of the proof for the Th.1 A classical representation of a Toeplitz determinant: D n (f ϕ ) = n 1 k=0 κ 2 k, Here p k (z) = κ k z k +... are polynomials, orthonormal on the unit circle w.r.t. f ϕ (θ). d dϕ ln D n(f ϕ ) = in terms of p n 1 (e ±iϕ ), p n (e ±iϕ ).

15 sketch of the proof for the Th.1 A classical representation of a Toeplitz determinant: D n (f ϕ ) = n 1 k=0 κ 2 k, Here p k (z) = κ k z k +... are polynomials, orthonormal on the unit circle w.r.t. f ϕ (θ). d dϕ ln D n(f ϕ ) = in terms of p n 1 (e ±iϕ ), p n (e ±iϕ ). We find the asymptotics of p n (z) by solving the associated Riemann- Hilbert problem (RHP). d dϕ ln D n(f ϕ ) = 1 2 n2 tan ϕ 2 n ( 2α tan ϕ 2 2 2α ) cos ϕ 1 cos ϕ/2 2 8 sin ϕ/2 ( + β2 cos ϕ/2 2 sin ϕ/2 α2 tan ϕ ) ( ) 2 1 cos ϕ/ O cos ϕ/2 2 sin ϕ/2 n sin 2, (ϕ/2) where the remainder term is uniform for 2s/n ϕ π ε, ε > 0.

16 sketch of the proof for the Th.1 D n (f ϕ )? as ϕ π from below. D n (f ϕ ) = 1 (2π) n n! 2π ϕ ϕ... 2π ϕ ϕ 1 j<k n n e iθj e iθ k 2 f(e iθj )dθ j. j=1

17 sketch of the proof for the Th.1 D n (f ϕ )? as ϕ π from below. D n (f ϕ ) = 1 (2π) n n! 2π ϕ ϕ... 2π ϕ After a change of variables we obtain: where ϕ 1 j<k n n e iθj e iθ k 2 f(e iθj )dθ j. j=1 D n (f ϕ ) = εn2 2 2αn (2π) n ( An + O(ε 2 ) ), ϕ = π ε, ε 0, A n = 1 n! i<j n n (z i z j ) 2 dz j j=1 Selberg integral The asymptotics of A n as n are known (Widom). Then, for ϕ = π ε ln D n (f ϕ ) = n 2 (ln ε ln 2)+2αn ln where O n (ε) 0, as ε 0. ln n ln 2+3ζ ( 1)+O n (ε),

18 sketch of the proof for the Th.1 D n (f ϕ )? as ϕ π from below. D n (f ϕ ) = 1 (2π) n n! 2π ϕ ϕ... 2π ϕ After a change of variables we obtain: where ϕ 1 j<k n n e iθj e iθ k 2 f(e iθj )dθ j. j=1 D n (f ϕ ) = εn2 2 2αn (2π) n ( An + O(ε 2 ) ), ϕ = π ε, ε 0, A n = 1 n! i<j n n (z i z j ) 2 dz j j=1 Selberg integral The asymptotics of A n as n are known (Widom). Then, for ϕ = π ε ln D n (f ϕ ) = n 2 (ln ε ln 2)+2αn ln where O n (ε) 0, as ε 0. ln n ln 2+3ζ ( 1)+O n (ε),

19 sketch of the proof for the Th.1 By integrating d dϕ ln D n(f ϕ ) and using the expression for D n (f ϕ ) as ϕ π, we obtain for any 2s n ϕ π ε: ln D n (f ϕ ) = (n 2 +2nα+2α 2 ) ln cos ϕ 2 +2(nα+α2 ) ln 1 + sin ϕ 2 cos ϕ ln n ( ) 1 4 β2 + α 2 ln sin ϕ 2 2α2 ln 2+ 1 ( ) 1 12 ln 2+3ζ ( 1)+O n sin ϕ. 2 Setting ϕ = 2s n, and using we obtain Theorem 1. D n (f 2s n P s = lim n D n (f) ),

20 sketch of the proof for the Th.2 Bessel kernel can be obtained as a scaling limit at the endpoint for the polynomials orthogonal on the interval [ 1, 1] that are related to the polynomials orthogonal on the unit circle with Fisher-Hartwig weight for β = 0. Consider the Hankel determinant with symbol ω(x): Dn H (ω) = det 1 1 x j+k ω(x)dx n 1 j,k=0.

21 sketch of the proof for the Th.2 Bessel kernel can be obtained as a scaling limit at the endpoint for the polynomials orthogonal on the interval [ 1, 1] that are related to the polynomials orthogonal on the unit circle with Fisher-Hartwig weight for β = 0. Consider the Hankel determinant with symbol ω(x): Dn H (ω) = det The Fredholm determinant P Bes s P Bes s 1 1 x j+k ω(x)dx = det(i K Bes s ): n 1 j,k=0 Dn H (ω 2s = lim ) n n Dn H (ω), ω ϕ(x) = f ϕ(e iθ ), x = cos θ. sin(θ).

22 sketch of the proof for the Th.2 Bessel kernel can be obtained as a scaling limit at the endpoint for the polynomials orthogonal on the interval [ 1, 1] that are related to the polynomials orthogonal on the unit circle with Fisher-Hartwig weight for β = 0. Consider the Hankel determinant with symbol ω(x): Dn H (ω) = det The Fredholm determinant P Bes s P Bes s 1 1 x j+k ω(x)dx = det(i K Bes s ): n 1 j,k=0 Dn H (ω 2s = lim ) n n Dn H (ω), ω ϕ(x) = f ϕ(e iθ ) sin(θ)., x = cos θ. Connection formula between Toeplitz and Hankel determinants: ( D H n (ω) ) 2 = π 2n 2 2(n 1)2 (κ 2n + p 2n (0)) 2 p 2n (1)p 2n ( 1) D 2n(f(z)), (Deift, Its, Krasovsky) here p n are polynomials orthonormal on the unit circle with the weight f(z).

23 G. Akemann, P.H. Damgaard, U. Magnea, and S. Nishigaki: Universality of random matrices in the microscopic limit and the Dirac operator spectrum, Nucl. Phys. B 487, no. 3, (1997), A.Borodin and G. Olshanski: Infinite random matrices and ergodic measures, Comm. Math. Phys. 223 (2001). no.1, J. des Cloizeaux and M. L. Mehta. Asymptotic behaviour of spacing distributions for the eigenvalues of random matrices, J. Math. Phys. 14, (1973) P. Deift, A. Its, and I. Krasovsky: Toeplitz and Hankel determinants with Fisher-Hartwig singularities P. Deift, A.Its, and I.Krasovsky: Asymptotics of the Airy-kernel determinant. Comm. Math. Phys. 278, (2008) P. Deift, A. Its, and X. Zhou. A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. 146, (1997)

24 F. Dyson: Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47, (1976) T.Erhardt: Dyson s constant in the asymptotics of the Fredholm determinant of the sine kernel. Comm. Math. Phys. 272, (2007) I.V.Krasovsky: Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. Int. Math. Res. Not. 2004, (2004) T.Nagao, K.Slevin: Nonuniversal correlations of random matrix ensembles. J. Math. Phys., 34 (1993), pp C.A. Tracy, H. Widom: Level Spacing Distributions and the Bessel kernel. Comm. Math. Ph. 161, (1994) H. Widom: The asymptotics of a continuous analogue of orthogonal polynomials. J. Approx. Th. 77, (1994) N.S. Witte, P.J. Forrester, Gap probabilities in the finite and scaled Cauchy random matrix ensembles, Nonlinearity 13 (200), no. 6,

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