Non-commutative Painlevé equations and Hermite-type matrix orthogonal polynomials

Non-commutative Painlev´e equations and Hermite-type matrix orthogonal polynomials ∗ Mattia Cafasso†and Manuel D. de la Iglesia‡ †LUNAM Universit´e, LAREMA, Universit´e d’Angers 2 Boulevard Lavoisier, 49100 Angers, France. [email protected] ‡Departamento de An´alisis Matem´atico. Universidad de Sevilla Apdo (P. O. BOX) 1160. 41080 Sevilla, Spain. [email protected] Abstract We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painlev´e IV differential equation for each family. 1 Introduction Let us denote with (pn)n∈Nthe classical Hermite polynomials such that deg(pn) = nand ZR pn(x)pm(x)e−x2dx =δnm. As we know from the pioneering work of Gaudin and Mehta [17], the so-called Hermite kernel Kn(x, y) := n−1 X k=0 pk(x)pk(y)e−x2+y2 2, describes the statistical properties of the eigenvalues of a random matrix in the space of (n×n) Hermitian matrices equipped with the measure µ(M) := e−Tr(M2)dM, where dM denotes the standard Haar measure. More precisely, µ(M) induces a measure on the space of configurations ∗The work of the first author is partially supported by the ANR grant DIADEMS, the work of the second author is partially supported by MTM2012-36732-C03-03 (Ministerio de Economia y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andalucia) and Feder Funds (European Union). We are also grateful to the Research Group on Orthogonal Polynomials and Approximation Theory of the University of Sevilla and to the G´eanpyl project, for financing visits to Sevilla and Angers. 1 arXiv:1301.2116v2 [math-ph] 13 May 2013 of npoints on the real line, hence a determinantal point process whose particles are given by the eigenvalues of M, and whose correlation functions ρk(x1, . . . , xk) are given by the formula ρk(x1, . . . , xk) = det(Kn(xi, xj))k i,j=1. In particular, the last particle distribution F(s), describing the probability that the largest eigenvalue is smaller than s, is given by the Fredholm determinant F(s) = det(Id −χsKn),where we denoted with χsthe indicator function of the semi infinite interval [s, ∞), and Knis the integral operator whose kernel is Kn(x, y) (for a very nice introduction about determinantal random point processes and random matrices see [16]). A remarkable connection between F(s) and the Painlev´e IV equation has been discovered in the nineties by Tracy and Widom. Namely, in [22], they proved that the log-derivative R(s) := ∂slog(F(s)) solves the sigma-form of Painlev´e IV equation (R00)2+ 4(R0)2(R0+ 2n)−4(sR0−R)2= 0. The aim of this article is to extend this result to the case of Christoffel-Darboux kernels associated to Hermite-type matrix-valued orthogonal polynomials (MOP). To that purpose we will show first how to obtain double integral representations of some examples of MOP. First, we need to introduce some notations. In what follows we consider Nfixed and we use preferably boldface letters to denote matrices, and standard font for scalars. We also use IN for the (N×N) identity matrix, omitting the explicit reference to its dimension when it cannot lead anyone into confusion. Let Cbe any piecewise smooth oriented curve. A weight matrix W= (Wij)N i,j=1 :C → GL(N, R) on the curve Cis a positive definite matrix at any point of the curve with finite moments. We say that a matrix function Fbelongs to the space L2 W(C,RN×N) if ZC F(z)W(z)FT(z)dz < ∞, where the superscript “T” denotes, as usual, the transpose1. In the above definition we mean that the integral is finite entry by entry. In the case when the weight matrix Wis the identity matrix INwe will just write L2C,RN×N. This induces a matrix-valued inner product for any two matrix functions F,G∈L2 W(C,RN×N), denoted by hF,GiW=ZC F(z)W(z)GT(z)dz. (1.1) This is not an inner product in the common sense, but it has properties similar to the usual scalar inner products. It is also possible to define a scalar norm of a matrix function Fby Tr (hF,FiW)1/2 (see [7]). Therefore L2 W(C,RN×N) with this norm is a Hilbert space and (1.1) is the inner product. A sequence (Pn)n∈Nof orthonormal MOP with respect to a weight matrix Wis a sequence of matrix polynomials satisfying deg Pn=n, hPn,PmiW=INδnm,∀n, m ∈N. We will mainly work with MOP on the real line, therefore C=R. Work in the last few years has revealed a number of explicit families of MOP on the real line. In many cases they are joint 1More generally one could consider matrix-valued complex weights, and in this case you have to substitute the transpose with the Hermitian conjugate. 2 eigenfunctions of some fixed differential operator with matrix coefficients independent of the degree nof the polynomials. This study was initiated in [9], but nontrivial examples had to wait until [10, 12]. These examples are the matrix analogue of the classical families of Hermite, Laguerre and Jacobi polynomials. Given a complete orthonormal family of MOP (Pn)n∈Nin L2 W(R,RN×N) the Christoffel-Darboux (CD) kernel is defined as (see, for instance, (2.26) of [13]) Kn(x, y) := n−1 X k=0 PT k(y)Pk(x), x, y ∈R.(1.2) We observe immediately the following properties: 1. Kn(x, y) = KT n(y, x). 2. ∀F∈L2 WR,RN×N,F(y) = hF(x),Kn(x, y)iW(reproducing kernel property). 3. Kn(x, z) = hKT n(z, y),Kn(y, x)iW. Observe, in particular, from the second equation, that we are thinking about the kernel as an integral operator acting on the left2for functions in L2 WR,RN×N. In Section 2 we find double integral representations of the CD kernel (1.2) (or rather a slight modification of (1.2)) for two Hermite-type families of MOP, already introduced in [10]. We first find integral representations of the families (Pn)n∈Nusing the corresponding second-order differential equation that they satisfy. To the best of our knowledge, this is the first time that integral representations of families of MOP are studied in detail. Secondly, in Section 3, we study the Fredholm determinant det(Id −χsKn) of the integral operator Knwith kernel Kn(x, y). Our main tool will be the theory of integrable operators `a la Its-Izergin-Korepin-Slavnov; see [15] and also [8] for a survey of the remarkable properties of these operators and applications to statistical mechanics, random matrices and orthogonal polynomials. More specifically, the Fredholm determinant det(Id −χsKn) will be identified with the isomonodromic tau function associated to a specific Riemann-Hilbert problem (see Appendix A and B). This Riemann-Hilbert problem, through a standard procedure, can be reduced to one with constant jumps, leading to a certain Lax system of equations. The compatibility conditions of the Lax system give a couple of matrix ordinary differential equations. Combining these two equations we obtain a non-commutative version of the derived nonlinear Painlev´e IV differential equation for each family, see Theorems 3.3 and 3.4. The contents of this section are very close, in spirit, to the one obtained by one of the authors and Marco Bertola in [6]. Finally, in Section 3.1.1, we give a symmetric formulation of the non-commutative Painlev´e IV equation. The non-commutative Painlev´e II equation used in [6] has been introduced by Retakh and Rubtsov in [19], where the authors obtained this equation as a reduction of a non-commutative analogue of Toda equations. Hence it appears desirable to verify if also the non-commutative Painlev´e IV equations obtained in this article can be written, analogously, as reductions of some suitable non-commutative Toda-type equations. The content in Section 3.1.1 goes in this direction. 2It is possible to work with a CD kernel acting on the right, but in that case we have to consider a different inner product defined by (F,G)W:= RRGT(x)W(x)F(x)dx. Now the CD kernel will be defined by KT n(x, y) := Pn−1 k=0 PT k(x)Pk(y) = Kn(y, x). 3 2 Hermite-type MOP and related CD kernels In this section we consider a couple of examples of MOP already introduced in [10]. Both are orthogonal with respect to Hermite-type weight matrices of the form W(x) := e−x2T(x)TT(x), x ∈R, where Tis certain matrix polynomial. Consider an orthonormal family of MOP (Pn)n∈N. Now define, for every n∈N, the orthonormal function Φn(x) := e−x2/2Pn(x)T(x). (Φn)n∈Nis a family of matrix functions orthonormal with respect to the identity matrix, i.e. hΦn,ΦmiIN=INδnm,∀n, m ∈N, where the inner product h·,·i is defined by (1.1). In our examples the family (Φn)n∈Nwill always be complete in the space L2R,RN×N(see Section 6 of [14] for more details). The Hermite-type CD kernel is then defined, slightly modifying (1.2), as3 Kn(x, y) := n−1 X k=0 ΦT k(y)Φk(x).(2.1) In order to find double integral representations of these two examples of matrix-valued kernels (2.1) we will use the fact that the corresponding MOP are eigenfunctions of a second-order differential equation of the form P00 n(x)F2(x) + P0 n(x)F1(x) + Pn(x)F0=ΓnPn(x),(2.2) where F2,F1and F0are matrix polynomials (which do not depend on n) of degrees less than or equal to 2, 1 and 0, respectively, and the eigenvalue Γnis a symmetric matrix. 2.1 The first example Let ANbe the (N×N) nilpotent (AN N=0) matrix AN:= N−1 X i=1 νiEi,i+1, νi∈R,(2.3) where Ei,j = (δirδsj)N r,s=1 is the elementary matrix with 1 at entry (i, j) and 0 elsewhere, and JN the diagonal matrix JN:= N X i=1 (N−i)Ei,i.(2.4) 3The kernel defined in this section, indeed, is equal to the one defined in the introduction up to a conjugation by e−x2 2T(x). In order to keep the notation simple we used, nevertheless, the same symbol. 4 Again, we will sometimes remove the dependence of Nand write A=ANand J=JN, whenever there is no confusion about the dimension of the matrices. Aand Jsatisfy the algebraic relation [A,J] = −A. Let Wbe the following weight matrix W(x) = e−x2eAxeATx, x ∈R,(2.5) already introduced in [10]. Observe that eAxis an upper triangular matrix polynomial of degree N−1 (since AN=0). The family of MOP Pn(x)=e−A2/4b Pn(x),(2.6) where ( b Pn)n∈Ndenotes the monic orthogonal family with respect to (2.5), satisfies a second-order differential equation as in (2.2) (see, for instance, Section 4 of [14]) where F2(x) = I,F1(x) = −2xI+ 2A,F0(x) = A2−2J,Γn=−2nI−2J. The family (2.6) is not orthonormal, but it will be normalized later for the computation of the CD kernel (2.1). In the following, given a matrix M, we will use the standard notation zM:= eMlog z, and the branch cut of log zis chosen to be the real negative axis. Theorem 2.1 Let (Pn)n∈Nbe the family of MOP defined by (2.6). Then there exist suitable constant matrices Cnand Dnsuch that Pn(x)eAx=Iγ z−JCnzJe−z2+2zx dz zn+1 ,(2.7) and Pn(x)eAx= ex2ZI wJDnw−Jew2−2xwwndw, (2.8) where the contour γencloses the origin, closes at −∞ and it is traversed in a counterclockwise direction while I:= L+iR, and L > 0is chosen so to have no intersection between γand I(see Figure 1). Proof: For the first integral representation (2.7) we observe, using the formula e−AxJeAx=J+xA, that the functions Yn(x) = Pn(x)eAxsatisfy the following differential equation Y00 n(x)−2xY0 n(x)−2Yn(x)J+ 2(nI+J)Yn(x) = 0.(2.9) Let us look for solutions of (2.9) of the form Yn(x) = Zη Vn(z)e2zxdz, where ηis some contour in the z-plane. Substituting this expression into the differential equation (2.9) and integrating by parts, it is easy to see that (2.9) holds if the two following conditions are satisfied V0 n(z) = −2z+n+ 1 zVn(z) + 1 z[Vn(z)J−JVn(z)] ,(2.10) zVn(z)e2zxη =0.(2.11) 5 I γ L Figure 1: Contours γand I. General solutions of homogeneous first-order matrix differential equations of the form X0(z) = A(z)X(z) + X(z)B(z) are easy to obtain considering solutions X(z) = X1(z)CX2(z), where Cis any constant matrix, such that X0 1(z) = A(z)X1(z) and X0 2(z) = X2(z)B(z) (see for instance [23], Lemma 30.1). Applying this to the equation (2.10), there exist, for every n≥0, constant matrices Cnsuch that the general solution is of the form Vn(z) = Vn,1(z)CnVn,2(z), where V0 n,1(z) = −2zI+n+ 1 zI+J zVn,1(z),V0 n,2(z) = Vn,2(z)J z. Therefore Vn(z) = z−JCnzJe−z2 zn+1 . gives a solution for (2.10). Now we observe that, choosing η=γ(the same contour as in the case of scalar Hermite polynomials), the expression on the left hand side of (2.11) is nothing but the residue at infinity of the expression zVn(z)e2xz, which is clearly zero because of the term e−z2. Hence Yn(x) := Iγ z−JCnzJe−z2+2zx dz zn+1 is a solution of (2.9). In particular, since Pn(x)eAxsatisfies a differential equation as in (2.9), there will be suitable constant matrices Cn, which depend on the family (Pn)n∈N, such that we get the integral representation (2.7). For the second integral representation (2.8) we follow a similar argument, just observing that the functions Zn(x)=e−x2Pn(x)eAxsatisfy the differential equation Z00 n(x)+2xZ0 n(x)−2Zn(x)J+ 2((n+ 1)I+J)Zn(x) = 0,(2.12) and considering Fourier type solutions of (2.12) of the form Zn(x) = Z∞ −∞ Vn(t)e2ixtdt. Then (as before) we find some first-order matrix differential equation for Vn(z) and we conclude that Vn(t) = tJ˜ Cnt−Je−t2tn, where ˜ Cn, n ≥0,are certain constant matrices. Hence Zn(x) = Z∞ −∞ tJ˜ Cnt−Je−t2+2ixttndt. 6 In particular, since e−x2Pn(x)eAxsatisfies a differential equation as in (2.12), by the change of variables t→iw, followed by a shift iw →iw +L, we conclude that there will be suitable constant matrices Dn=in+1iJ˜ Cni−J, depending on the family (Pn)n∈N, such that (2.8) holds. Q.E.D Remark 2.1 Observe that the constant matrices Cnand Dnobtained initially from the differential equations (2.9) and (2.12), respectively, have many degrees of freedom. It is exactly the choice of the family (Pn)n∈Nwhich allows us to have a unique representation of Cnand Dn, as we will see below. The meaning of this is because there are many families of matrix polynomials satisfying (2.9) or (2.12), but not all of them are orthogonal with respect to some weight matrix. Corollary 2.1 The Hermite-type CD kernel (2.1) associated with the MOP with weight (2.5) can be written as Kn(x, y)=e(x2−y2)/2ZI dw Iγ dzMn(z, w)ew2−2xw−z2+2zy,(2.13) where Mn(z, w) := zJ"n−1 X k=0 CT kz−JkPkk−2 WwJDkw zk#1 zw−J,(2.14) and kPkk2 Wdenotes the matrix-valued norm with respect to (1.1). Proof: It is just enough to use the formula Φn(x)=e−x2/2kPnk−1 WPn(x)eAx and then the integral representations (2.7) (with the substitution x→y) and (2.8) in the definition of the matrix-valued kernel (2.1). Q.E.D The coefficients Cnand Dnare not easy to obtain for our example since there are no general structural formulas like the norms or the coefficients of our family of MOP for any general size N. We will study in detail the case N= 2 in the next subsection. 2.1.1 A detailed study of the case N= 2 Let us consider the (2 ×2) case related to MOP with respect to the weight W(x) = e−x2eA2xeAT 2x,A2:= 0ν 0 0 , ν ∈R. We have many structural formulas for this example (see [11]). In particular, the polynomials (Pn)n∈Nin (2.6) have diagonal norms kPnk2 W=n!√π 2nγ2 n+1 0 0 1/γ2 n,(2.15) where we denoted γ2 n:= 1 + n 2ν2. The coefficients Cnand Dnof the integral representations (2.7) and (2.8) are given by Cn=n! 2n+1πi 1ν(n+ 1)/2 −ν/γ2 n1/γ2 n,(2.16) 7 and Dn=1 i√π 1ν −nν 2γ2 n 1/γ2 n!.(2.17) These coefficients are computed directly from the integral representations (2.7) and (2.8), since we have that the family (Pn)n∈Ncan be given in terms of scalar Hermite polynomials (see [11]). Therefore (2.16) is obtained from (2.7) using Cauchy’s residue theorem and (2.17) is obtained from (2.8) using standard manipulations of the Fourier transform of Hermite polynomials. Proposition 2.1 The Hermite-type CD kernel (2.13), for N= 2, can be written as Kn(x, y) = 2 (2πi)2e(x2−y2)/2ZI dw Iγ dz zJ2Bnz−J2wJ2B−1 nw−J2ew2−2xw−z2+2zy+nlog(w/z) w−z, (2.18) where Bn:= 1−ν nν 21!,det(Bn) = γ2 n.(2.19) Proof: Computing (2.14), using (2.15), (2.16) and (2.17), we have that Mn(z, w) = 2 (2πi)2           −1 w−z+w zn w−zz(γ2 n−1) + w wγ2 nν  w zn γ2 n−1   nν 2γ2 nw zn zw −1 w−z+w zn w−zw(γ2 n−1) + z zγ2 n           . Then, using Cauchy’s residue theorem, we observe that the diagonal terms −1/(w−z) and the −ν in the upper right corner of Mn(z, w) do not give any contribution. Hence we have Kn(x, y) = 2 (2πi)2e(x2−y2)/2ZI dw Iγ dz ˜ Mn(z, w)ew2−2xw−z2+2zy+nlog(w/z) w−z where ˜ Mn(z, w) =       z(γ2 n−1) + w wγ2 n ν γ2 n (w−z) nν 2γ2 n (z−1−w−1)w(γ2 n−1) + z zγ2 n      , and finally it is easy to see that ˜ Mn(z, w) = zJ2Bnz−J2wJ2B−1 nw−J2, where Bnis given by (2.19). Q.E.D 8 Remark 2.2 Also for the case N= 3, the Hermite-type CD kernel (2.13) has a representation analogue to (2.18), for a given (3×3) constant matrix Bn(we do not report here the computations, since they are completely analogues but more cumbersome). We conjecture that for any N≥1and a given n≥0, there always exists a constant matrix Bnsuch that Kn(x, y)= 2 (2πi)2e(x2−y2)/2 ZI dw Iγ dz zJNBnz−JNwJNB−1 nw−JNew2−2xw−z2+2zy+nlog(w/z) w−z.(2.20) 2.2 The second example Let Wbe the following weight matrix W(x)=e−x2eBx2eBTx2, x ∈R,(2.21) where B=A(I+A)−1and Ais the (N×N) nilpotent matrix (2.3). The family of MOP Pn(x) = [(I+A)−1/2]2n+1 b Pn(x), n ∈N,(2.22) where ( b Pn)n∈Ndenotes the monic orthogonal family with respect to (2.21), satisfies a second-order differential equation as in (2.2) (see Section 5 of [14]) with F2(x) = I,F1(x)=2x(2B−I),F0(x) = 2(B−2J),Γn=−2nI−4J, where Jis the diagonal matrix (2.4). Also in this case, in complete analogy with the first example, we can deduce two different integrable representations of the polynomials (Pn)n∈Nand, consequently, an integral representation of the related Hermite-type CD kernel. Theorem 2.2 Let (Pn)n∈Nbe the family of MOP defined by (2.22). Then there exist suitable constant matrices Cnand Dnsuch that Pn(x)eBx2=Iγ z−2JCnz2Je−z2+2zx dz zn+1 ,(2.23) and Pn(x)eBx2= ex2ZI w2JDnw−2Jew2−2xwwndw, (2.24) where the contour γencloses the origin, closes at −∞ and it is traversed in a counterclockwise direction while I:= L+iR, and L > 0is chosen so to have no intersection between γand I(see again Figure 1). Proof: This theorem is proven exactly in the same way as Theorem 2.1, using as starting point the formula e−Bx2JeBx2=J+x2(B−B2) instead of e−AxJeAx=J+xA.Q.E.D Corollary 2.2 The Hermite-type CD kernel (2.1) can be written as Kn(x, y)=e(x2−y2)/2ZI dw Iγ dzMn(z, w)ew2−2xw−z2+2zy,(2.25) where Mn(z, w) = z2J"n−1 X k=0 CT kz−2JkPkk−2 Ww2JDkw zk#1 zw−2J.(2.26) and kPkk2 Wdenotes the matrix-valued norm with respect to (1.1). 9 The compatibility conditions give the following coupled system of ODEs:    u0=−u2+ 2su+ 4z−2nI2+VB, z00 = 2u0z+ 2uz0−2sz0, where VB:= 4J2−2yJ3y†. Combining these two equations we obtain a non-commutative version of the derived PIV equation, in the form u000+ [u00,u]−4(n+1+s2)u0−2{u0,u2}+uu0u +6s{u0,u}+ 4u(u−sI2)+(V0 B−2(uVB))0+ 2sV0 B=0. (3.15) Remark 3.1 Before going into the proof let us remark that both the equations (3.13) and (3.15) are equations for just one variable, namely y;uand V(either VAor VB) being functions of y. The reason why we claim that this equation is a non-commutative version of the derived PIV is that, in both cases, if we assume that all the variables commute, we get the equation u000 −4u0−6u2u0+ 12u0u−4nu0+ 4u2−4su −4s2u0= 0, and the reader can easily verify that this latter equation is the derivative of the standard PIV equation u00 =(u0)2 2u+3 2u3−4su2+ 2(s2+1+n)u−2n2 u. Proof of Theorem 3.3: Given Γ(λ) solution of the Riemann-Hilbert problem 3.1 with G(λ) as in (3.9), we have Zγ∪I Tr Γ−1 −(λ)∂λΓ−(λ)Ξ(λ)dλ 2πi =−res λ=∞Tr Γ−1 −(λ)∂λΓ−(λ)∂sTA,(3.16) where the (formal) residue above simply stands for minus the coefficient of the power λ−1in the asymptotic expansion of the argument. The formula (3.16) can be proven using Cauchy theorem and goes back to the article of Palmer [18]. A very precise detailed derivation is given in [4], section 5.1. Direct application of the formula (3.16) and the equation (3.7) yields the equation (3.10). Next we observe that Ψ(λ) = Γ(λ)eTA(λ)solves a Riemann-Hilbert problem with constant jump, hence both (∂λΨ)Ψ−1and (∂sΨ)Ψ−1are meromorphic functions on C∗so that, in particular, considering the singular behavior at 0 at ∞, we get    (∂λΨ)Ψ−1=A=λA1+A0+λ−1A−1 (∂sΨ)Ψ−1=U=λU1+U0. Let us start with A; in order to compute the coefficients A1,A0,we compute the first terms of the asymptotic expansion at infinity of (∂λΨ)Ψ−1giving immediately5 A1=I2⊗σ3,A0=−sI2⊗σ3+ [Γ1,I2⊗σ3] = −sI2⊗σ3+0−2(Γ1)12 2(Γ1)21 0. 5We denote with σ3the standard Pauli matrix σ3= diag(1,−1). 16 The equation for A−1is slightly more complicated: we use again the expansion at infinity and we get A−1= [Γ2,I2⊗σ3]+[I2⊗σ3,Γ1]Γ1+n 2I2⊗σ3−J2⊗I2−[Γ1, sI2⊗σ3].(3.17) In order to simplify the expression above we use the fact that the λ−1-term of (∂sΨ)Ψ−1term is identically zero, giving Γ0 1= [Γ2,I2⊗σ3]+[I2⊗σ3,Γ1]Γ1,(3.18) leading eventually to A−1=Γ0 1−[Γ1, sI2⊗σ3] + n 2I2⊗σ3−J2⊗I2. The block-diagonal part of the equation above gives (Γ1)0 11 = 2(Γ1)12(Γ1)21,(Γ1)0 22 =−2(Γ1)21(Γ1)12,(3.19) and so, combining (3.17), (3.18) with the definitions (3.11) and using the relations (3.19) we get A0= −sI2y 2y−1zsI2 ,A−1=    −z+n 2I2−J2−uy 2 (A−1)21 y−1zy −n 2I2−J2    . In order to get the missing term (A−1)21 we use the Lax equation ∂sA−∂λU= [U,A]; the (1,1) term giving (A−1)21 =y−1z0−y−1uz. In this way we arrive to the stated expression for A. The computation for U, since we have no singularities at the origin, is simpler and the asymptotic at infinity of (∂sΨ)Ψ−1gives immediately U1=−I2⊗σ3,U0=  02(Γ1)12 −2(Γ1)21 0 =  0−y −2y−1z0 . The first equation in (3.12) is just the (1,2) (block) entry of the equation. For the second one we start observing that the off-diagonal terms of the relation (3.18) gives the equations (Γ1)0 12 =−2(Γ2)12 + 2(Γ1)12(Γ1)22,(Γ1)0 21 = 2(Γ2)21 −2(Γ1)21(Γ1)11.(3.20) Then we go on analyzing the asymptotic expansion at infinity of (∂λΨ)Ψ−1; the λ−2-term (which is identically zero) gives the equation Γ1=[Γ3,I2⊗σ3]+[sI2⊗σ3,Γ2]+hΓ1,n 2I2⊗σ3−J2⊗I2i +[Γ1, sI2⊗σ3]Γ1+[Γ1,I2⊗σ3]Γ2 1+[I2⊗σ3,Γ1]Γ2+[I2⊗σ3,Γ2]Γ1. (3.21) 17 The (1,1)-entry of this equation gives (Γ1)11 =−2s(Γ1)12(Γ1)21 −2(Γ1)12(Γ1)21(Γ1)11 −2(Γ1)12(Γ1)22(Γ1)21 + 2(Γ1)12(Γ2)21 + 2(Γ2)12(Γ1)21, (3.22) and combining (3.20) with (3.22) we get the second equation in (3.12) (the computation is lengthy but completely straightforward). The equation (3.13) is obtained simply expressing zin function of uusing the first equation in (3.12) and then substituting in the second one. Q.E.D Since the proof of the Theorem 3.4 is formally identical to the one of Theorem 3.3 we will not write it here. One has just to substitute J2with J3where appropriate, and substitute y−1with y†. Indeed, going through the computations of the proof above, is easy to verify that we use just the property yy−1=I2, while y−1y=I2is never used. 3.1.1 A symmetric formulation of PIV While formulas (3.13), (3.15) establish a direct connection with the classical PIV equation and Tracy-Widom results [22], in view of equations (3.10) and (3.14) it would be desirable to write a system of ODEs in which both the entries (Γ1)11 and (Γ1)22 appear explicitly as dependent variables. To this aim the symmetric formulation of PIV given (in the scalar case) by Aratyn, Gomes and Zimerman in [2, 3] is particularly suitable. A similar description can be given in our non-commutative case. We introduce new notations for the entries of the Lax matrices so to match with the cited articles. Theorem 3.5 Let Ψ(λ) := Γ(λ)eTA(λ)be as in Theorem 3.3 and denote                    q:= 2(Γ1)12, r:= −2(Γ1)21, ρR:= 4(Γ1)11, ρL:= −4(Γ1)22. Then the related Lax matrices Aand Ureads U=−λI2q rλI2, A=(λ−s)I2−q −r−(λ−s)I2+1 4λρ0 R+ 2nI2−4J24sq+ 2q0 4sr−2r0−ρ0 L−2nI2−4J2, 18 while Lax equations read                        ρR= 2sqr +q0r−qr0, ρL= 2srq +rq0−r0q, −sq0+1 2(−q00 + 2qrq) = (1 + 2n)q+ 2[q,J2], sr0−1 2(r00 −2rqr)=(−1+2n)r+ 2[r,J2]. (3.23) Theorem 3.6 Let Ψ(λ) := Γ(λ)eTB(λ)be as in Theorem 3.4 and denote                    q:= 2(Γ1)12, r:= −2(Γ1)21, ρR:= 4(Γ1)11, ρL:= −4(Γ1)22. Then the related Lax matrices Aand Ureads U=−λI2q rλI3, A=(λ−s)I2−q −r−(λ−s)I3+1 4λρ0 R+ 2nI2−8J24sq+ 2q0 4sr−2r0−ρ0 L−2nI3−4J3, while Lax equations read                        ρR= 2sqr +q0r−qr0, ρL= 2srq +rq0−r0q, −sq0+1 2(−q00 + 2qrq) = (1 + 2n)q+ 4[q,J2], sr0−1 2(r00 −2rqr)=(−1+2n)r+ 2[r,J3]. (3.24) Remark 3.2 The system of equations above (3.23) and (3.24) are the non-commutative analogues of equations (2.4) in [2], which are equivalent to the sigma-form of PIV (see equations (2.6),(2.7) in [2]) Proof: Both theorems are proven in the same way and the proof consists, essentially, on rewriting the same equations as in Theorems 3.3 and 3.4 with different variables. In particular the third and the fourth equations in (3.23), (3.24) come from the the Lax equation ∂sA−∂λU= [U,A], 19 together with the already used fact that ρ0 R=−qr and ρ0 L=−rq (see equations (3.19)). The other two equations come from the (already used) equations (3.20) combined with (3.22) and (Γ1)22 = 2 (s(Γ1)21(Γ1)12 + (Γ1)21(Γ1)11(Γ1)12 + (Γ1)21(Γ1)12(Γ1)22 −(Γ1)21(Γ2)12 −(Γ2)21(Γ1)12), this latter coming from the (2,2)-entry of (3.21). Q.E.D 4 Concluding remarks In this paper we have shown (for the first time, to the best of our knowledge) some integral representations of two particular examples of Hermite-type MOP on the real line, and then deduced a double integral representation of the related Christoffel-Darboux kernel Kn(x, y). This was the starting point for the study of the Fredholm determinant det(Id −χsKn), where Knis the integral operator with kernel Kn(x, y). Using some Riemann-Hilbert techniques, we related this Fredholm determinant with a non-commutative version of the Painlev´e IV equation. We remark that the type of orthogonality we started from is essential in order to get that specific equation. In complete analogy with the scalar case [22], for instance, we expect that Laguerre-type matrix orthogonal polynomials should be related to some non-commutative version of the Painlev´e V equation. We plan to investigate on this issue in subsequent works. Another interesting question that arises is how these kernels Kn(x, y) behave if we consider scaling limit as n→ ∞. It is very well known that, after rescaling appropriately the variables, the Hermite kernel converges to the Airy kernel in the following way lim n→∞ 1 √2n1/6KHermite n√2n+x √2n1/6,√2n+y √2n1/6=KAi(x, y), where xand yare in a bounded set (see [1] for a very nice and elementary deduction of this convergence, even in a more general setting than what needed here). In the matrix case, for the two Hermite-type kernels we study in this paper (see (2.20) and (2.33)) there will be a scalar behavior as n→ ∞, i.e. lim n→∞ 1 √2n1/6Kn√2n+x √2n1/6,√2n+y √2n1/6=KAi(x, y)IN. In particular, this property holds for the examples (2.18) and (2.31) (N= 2). This is easy to see from the integral representation (2.20) (analogous for (2.33)) using classical steepest descent methods, as in [1]. Indeed the “matrix part” of the kernel can be written as zJNBnw zJNB−1 nw−JN, but under the given rescaling6w z→1 as n→ ∞, therefore the matrix part converges to the identity matrix INas n→ ∞, no matter the choice of Bn. The Airy kernel is related, as it is well known, to the Painlev´e II equation [21], and this relation extends to the matrix case for a specific type of matrix Airy kernel [6]. For the two Hermitetype polynomials we study in this paper, the scaling limit of the corresponding CD kernel has a scalar behavior, so that the non-commutativity disappears and the corresponding matrix Painlev´e II equation is nothing but Nnon–interacting copies of the scalar equation. It would be interesting to 6Recall that the variables wand zare rescaled as w=qn 21 + µ n1/3and z=qn 21 + λ n1/3, see [1]. 20 find other Hermite-type MOP whose kernels do not behave in this way, something that in principle is not an easy task. This consideration, however, goes beyond the scope of this paper, and will be pursued elsewhere. Acknowledgements: The first author is grateful to V. Retakh and V. Rubtsov for many interesting discussions about their works on non-commutative integrable systems. A A brief reminder of regularized Fredholm Determinants We refer to [20] for the relevant details: we shall need only the elementary facts which we recall here. In general the Fredholm determinant of an operator of the form (Id −G) can be defined only when Gis of trace class. Recall that if Gis represented as an integral operator on a (separable) Hilbert L2(X, dµ) with kernel G(x, y) (we abuse notation here) then det(Id −G) = 1 + ∞ X n=1 1 n!ZXn det[G(xi, xj)]i,j≤n n Y i=1 dµ(xi).(A.1) There are other trace ideals Ip,p∈N,which means that Gpis trace-class [20]; in particular I2 consists of Hilbert-Schmidt operators. For G∈ Ipone can define following Carleman a regularized determinant detp(Id −G) which has the same main property of vanishing iff the operator is not invertible. In particular for Hilbert-Schmidt operators one has det2(Id −G) := 1 + ∞ X n=1 1 n!ZXn det[G(xi, xj)(1 −δij)]i,j≤n n Y i=1 dµ(xi), that is, one simply omits the diagonal elements in the determinants under the integral sign. This determinant has the properties •if Gis also trace-class then det2(Id −G) = det(Id −G)e−trG. •if G1, G2are Hilbert-Schmidt operators (and hence G1G2is trace class) then det2(Id −G1)det2(Id −G2) = det2(Id −G1−G2+G1G2)e−tr(G1G2). An interesting occurrence (which is used in this article) is that if Gis just Hilbert-Schmidt but its kernel vanishes on the diagonal G(x, x)≡0 then the series defining det2(Id −G) is identical to the regular det(Id −G). The reason for still wanting to distinguish det2from det in this case is simply that Gmay fail to have a trace and in a different basis the ordinary det may simply be ill-defined. B Integrable kernels and isomonodromic tau functions In this appendix we recall some basic facts about integrable kernels `a la Its-Izergin-Korepin-Slavnov [15] and their connections with isomonodromic tau functions, recalling in particular a theorem 21 proved in [5] (see also [6]). Given a piecewise smooth oriented curve Con the complex plane (possibly extending to infinity) and two matrix-valued functions f,g:C −→ Matp×k(C), we define the kernel Kas K(λ, µ) := fT(λ)g(µ) λ−µ. We say that such kernel is integrable if fT(λ)g(λ) = 0(so that it is non-singular on the diagonal). We are interested in the operator K:L2(C,Ck)→L2(C,Ck) acting on k-vector functions via the formula (Kh)(λ) = ZC K(λ, µ)h(µ)dµ, and, in particular, we are interested in the Fredholm determinant det(Id −K) defined as in (A.1). The key observation is that, denoting with ∂the differentiation with respect to any auxiliary parameter on which Kmay depend, we obtain the formula ∂log det(Id −K) = −Tr((Id + R)∂K),(B.1) where Ris the resolvent operator, defined as R= (Id −K)−1K. Moreover Ris again an integrable operator, i.e. R(λ, µ) = FT(λ)G(µ) λ−µ, and F,Gcan be found solving the following RH problem:            Γ+(λ) = Γ−(λ)M(λ), λ ∈ C, Γ(λ) = I+O(λ−1), λ −→ ∞, M(λ) = I−2πif(λ)gT(λ). (B.2) More precisely we have the two equalities F(λ) = Γ(λ)f(λ),G(λ)=(Γ−1)T(λ)g(λ). Now suppose, as in the formula (B.1), that the operator K(and hence the Riemann-Hilbert problem (B.2) depends smoothly on a certain parameter set of parameters7. On the space of these deformation parameters, we introduce the following one-form (here below ∂denotes a vector in the space of deformation parameters) ωM(∂):= ZC TrΓ−1 −(λ)∂λΓ−(λ)Ξ∂(λ)dλ 2πi,(B.3) Ξ∂(λ):= ∂M(λ)M−1(λ) The definition (B.3) is posed for arbitrary jump matrices; in the case of the Riemann-Hilbert problem (B.2) the spontaneous question arises as to whether ωMin (B.3) and the Fredholm determinant are related. The answer is positive within a certain explicit correction term, as in the theorem below 7In the case treated in this article, we just have one parameter, namely s. 22 Theorem B.1 ([5]) 8Let f(λ;~s),g(λ;~s) : C×S−→ Matp×k(C)and consider the Riemann-Hilbert problem with jumps as in (B.2). Given any vector field ∂in the space of the parameters Sof the integrable kernel we have the equality ωM(∂) = ∂log det(Id −K) + H(M), where ωM(∂)is as in (B.3) and H(M) := H1(M)−H2(M) = ZC Tr∂f0Tg+f0T∂gdλ −2πi ZC Tr(gTf0∂gTf)dλ. 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