R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Pavel M. BLEHER & Arno B.J. KUIJLAARS Integral representations for multiple Hermite and multiple Laguerre polynomials Tome 55, no 6 (2005), p. 2001-2014. <http://aif.cedram.org/item?id=AIF_2005__55_6_2001_0> © Association des Annales de l’institut Fourier, 2005, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 55, 6 (2005), 2001–2014 INTEGRAL REPRESENTATIONS FOR MULTIPLE HERMITE AND MULTIPLE LAGUERRE POLYNOMIALS by Pavel M. BLEHER & Arno B.J. KUIJLAARS (*) 1. Multiple orthogonal polynomials. Multiple orthogonal polynomials are an extension of orthogonal poly- nomials that play a role in the random matrix ensemble with an external source 1 (1.1) e− Tr(V (M)−AM)dM Zn defined on n × n Hermitian matrices, see [5, 6, 13]. Here A is a fixed n × n Hermitian matrix and V : R → R is a function with enough increase at ±∞ so that the integral − Tr(V (M)−AM) Zn = e dM converges. Random matrices with external source were introduced and studied by Br´ezin and Hikami [8, 9, 10, 11, 12], and P. Zinn-Justin [21, 22]. Related recent work includes [2, 6, 15, 19]. (*) The first author was supported in part by NSF Grants DMS-9970625 and DMS- 0354962. The second author was supported by projects G.0176.02 and G.0455.04 of FWO-Flanders, by K.U. Leuven research grant OT/04/24, by INTAS Research Network NeCCA 03-51-6637, and by the European Science Foundation Program Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM) and the European Network in Geometry, Mathematical Physics and Applications (ENIGMA). Keywords: Multiple orthogonal polynomials, random matrices, Christoffel-Darboux for- mula. Math. classification: 42C05, 15A52. 2002 Pavel M. BLEHER & Arno B.J. KUIJLAARS In what follows, we assume that A has m distinct eigenvalues a1,...,am of multiplicities n1,...,nm. We consider m fixed and use multi- index notation n =(n1,...,nm) and | n| = n1 + ···+ nm. The average characteristic polynomial Pn(x)=E [det(I − M)] of the ensemble (1.1) is a monic polynomial of degree | n| which satisfies for k =1,...,m, ∞ j (1.2) Pn(x)x wk(x)dx =0,j=0,...,nk − 1, −∞ where −iew(V (x)−akx) (1.3) wk(x)=e , see [5]. The relations (1.2) characterize the polynomial Pn uniquely. For A = 0, we have that Pn is the usual orthogonal polynomial with respect to the weight e−(V (x)), which is a well-known fact from random matrix theory. For general m, the relations (1.2) are multiple orthogonality relations with respect to the weights (1.3) and the polynomial Pn is called a multiple orthogonal polynomial of type II. The multiple orthogonal polynomials of type I consist of a vector (1) (2) (m) (k) − (1.4) (An ,An ,...,An ), deg An nk 1, of polynomials such that the function m (k) (1.5) Qn(x)= An (x)wk(x) k=1 satisfies ∞ | |− j 0,j=0,..., n 2, (1.6) x Qn(x)dx = −∞ 1,j= | n|−1. (k) The polynomials An are uniquely determined by the degree requirements (1.4) and the type I orthogonality relations (1.6). By the Weyl integration formula, the random matrix ensemble (1.1) has the following joint eigenvalue distribution n 1 − ∗ e V (λj ) eAUΛU dU (λ − λ )2 dλ dλ ···dλ ˜ j k 1 2 n Zn j=1 j<k where dU is the normalized Haar measure on the unitary group U(n) and Λ = diag(λ1,...λn). Using the confluent form of the Harish-Chandra / ∗ Itzykson-Zuber formula [14, 16] to evaluate the integral eAUΛU dU,we find that the joint eigenvalue distribution can be written as 1 ··· (1.7) det (fj(λk))1j,kn det (gj(λk))1j,kn dλ1dλ2 dλn, Zn ANNALES DE L’INSTITUT FOURIER MULTIPLE HERMITE ANDLAGUERRE POLYNOMIALS 2003 j−1 where fj(x)=x and g1,...,gn are the functions j −(V (x)−akx) (1.8) x e ,k=1,...,m, j =0, 1,...,nk − 1 taken in some arbitrary order. Then (1.7) is a biorthogonal ensemble in the sense of Borodin [7], which in particular implies that the eigenvalue point process is determinantal, that is, there is a kernel K(x, y) such that the k point correlation functions have determinantal form (1.9) det (K(λi,λj))1i,jk . The multiple orthogonal polynomials of types II and I can be used to biorthogonalize the functions fj and gj and to give an explicit formula for K. Choose a sequence of multi-indices n0, n1,..., nn = n such that | nj| = j and nj nj+1 and define pj(x)=Pnj (x),qj(y)=Qnj+1 . Then pj+1 ∈ span{f1,...,fj}, and qj+1 ∈ span{g1,...,gj} for a suitable ordering of the functions (1.8). In addition we have the biorthogonality ∞ pi(x)qj(x)dx = δi,j. −∞ As in [7] it then follows that n−1 (1.10) K(x, y)= pj(x)qj(y). j=0 By the Christoffel-Darboux formula for multiple orthogonal polynomials [13] the kernel K satisfies m h(k) − − n (1.11) (x y)K(x, y)=Pn(x)Qn(y) Pn−ek (x)Qn+ek (y) h(k) k=1 n−ek where ∞ (k) nk (1.12) hn = Pn(x)x wk(x)dx −∞ m and ek is the kth standard basis vector in R . In the following two sections we study two special cases related to multiple Hermite polynomials and multiple Laguerre polynomials. These 1 2 cases correspond to the random matrix model (1.1) with V (M)= 2 M and V (M)=M respectively (in the latter case we restrict to positive definite matrices). Br´ezin and Hikami [9] and Baik, Ben Arous, and P´ech´e [4] gave double integral representations for the correlation kernels for these cases. We will derive integral representations for the multiple Hermite and multiple Laguerre polynomials, and use that to show that the kernels agree with the multiple orthogonal polynomial kernel (1.11). TOME 55 (2005), FASCICULE 6 2004 Pavel M. BLEHER & Arno B.J. KUIJLAARS 2. Multiple Hermite polynomials. 1 2 The special case V (M)= 2 M was considered in a series of papers of Br´ezin and Hikami, [8, 9, 10, 11, 12]. This case corresponds to M = H + A − 1 Tr H2 where H is a random matrix from the GUE ensemble (1/Zn)e 2 dH and A is fixed as before. In [9] the following expression for the kernel was derived i∞ m nk 1 1 − 2− 1 − 2 s−ak 1 2 (s x) 2 (t y) (2.1) K(x, y)= 2 ds dt e (2πi) − ∞ t−ak s−t i Γ k=1 where Γ is a closed contour encircling the points a1,...,am once in the positive direction, and the path from −i∞ to i∞ does not intersect Γ, see also Johansson [17]. 1 2 When V (x)= 2 x , the multiple orthogonal polynomials are called multiple Hermite polynomials, since they clearly generalize the usual Hermite polynomials [1, 3, 20]. We derive integral representations for the multiple Hermite polynomials of type I and type II, which resemble the integral representation (2.1) of the kernel. 2.1. Multiple Hermite polynomials of type II. The multiple Hermite polynomial Pn is the monic polynomial of 1 2 − x +akx degree | n| that satisfies (1.2) with wk(x)=e 2 , k =1,...,m. Theorem 2.1. — The multiple Hermite polynomials of type II has the integral representation i∞ m 1 1 − 2 √ 2 (s x) nk (2.2) Pn(x)= e (s − ak) ds. − ∞ 2πi i k=1 Proof. — Let us denote the left-hand side of (2.2) by P (x). After performing the change of variables s = t + x, we get i∞ m 1 1 2 t nk (2.3) P (x)=√ e 2 (t + x − ak) dt, − ∞ 2πi i k=1 which shows that P is a polynomial of degree | n| with leading coefficient i∞ 1 1 t2 √ e 2 dt =1. 2πi −i∞ ANNALES DE L’INSTITUT FOURIER MULTIPLE HERMITE ANDLAGUERRE POLYNOMIALS 2005 So P is a monic polynomial. Now we use (2.3) to compute for k =1,...,m, and j =0, 1,..., ∞ j − 1 x2+a x (2.4) P (x)x e 2 k dx −∞ 1 a2 i∞ ∞ m e 2 k 1 2 2 (t −(x−ak) ) j nl = √ e 2 x (t + x − al) dxdt. − ∞ −∞ 2πi i l=1 Switching to polar coordinates x − ak = r cos θ, t = ir sin θ, we find that the right-hand side of (2.4) is equal to 1 a2 ∞ 2π e 2 k 1 2 − r nk+1 j inkθ iθ nl (2.5) √ e 2 r (ak+r cos θ) e (re +ak−al) dθ dr. 2π 0 0 l= k The θ-integral vanishes for j =0,...,nk − 1, since the integrand can be ipθ written as a linear combination of e with integer p nk − j.