A fully noncommutative Painlev\'e II hierarchy: Lax pair and solutions related to Fredholm determinants

We consider Fredholm determinants of matrix convolution operators associated to matrix versions of the $n - $th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlev\'e II hierarchy, defined through a matrix valued version of the Lenard operators. In particular, the Riemann-Hilbert technique used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitely written in terms of these matrix valued Lenard operators and some solution of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy convolution operators.


Introduction
The aim of this work is to relate a family of solutions of a fully noncommutative version of the Painlevé II hierarchy to the Fredholm determinants of some matrix version of the n−th Airy convolution operators. The scalar version of these convolution operators have been recently studied in [15], in relation with determinantal point processes (we will discuss this relation later on). In order to construct the matrix analogue of these convolution operators, we first define a matrixvalued version of the n-th Airy function, in the following way Ai 2n+1 (x, s) = (c j,k Ai 2n+1 (x + s j + s k )) r j,k=1 , c j,k ∈ C, x ∈ R , (1.1) where Ai 2n+1 (x + s j + s k ) is a shift of the n−th scalar Airy function, for some real parameters s l , l = 1, . . . , r. We recall that the n−th scalar Airy function, Ai 2n+1 (x), is defined as a particular solution of the differential equation for each n ≥ 1. We refer to [16] for details about the soutions of the generalized Airy equations (1. for any f = (f 1 , . . . , f r ) T ∈ L 2 (R + , C r ). It is actually on the square of this sequence of operators that we focused our study, and in particular on the Fredholm determinants defined as F (n) (s 1 , . . . , s r ) := det Id R + − Ai 2 2n+1 , (1.4) that are well defined since the operators Ai 2 2n+1 are trace-class. The core of this work is indeed to establish a relation between these Fredholm determinants (1.4) and some solution of a fully noncommutative Painlevé II hierarchy. In particular, the results achieved in [3], where the authors extend the theory of integrable operators of Its-Izergin-Korepin-Slavnov ( [10]), can be directly applied to the matrix convolution operators Ai 2n+1 defined in (1.3). As by product, an equality between the Fredholm determinants F (n) (s 1 , . . . , s r ) and those of certain integrable operators can be established. The study of these integrable operators involves particular Riemann-Hilbert problems (defined in (2.1)), and these are indeed the main tool used in the following. In particular: starting from them we construct a solution for the isomonodromic Lax pair of the fully noncommutative Painlevé II hierarchy, that we are going to define. To start with, we first define a matrix version of the well known Lenard operators L n , used to define the scalar Painlevé II hierarchy. In the following, W is a matrix-valued function depending on all the parameters s l , l = 1, . . . , r with values in Mat(r × r, R), and in this sense is a noncommutative variable. Thus the symbols [ , ] and [ , ] + are needed, and they indicate respectively the standard commutator and anti-commutator between two matrices. Then each operator L n is defined by the following recursive relation This noncommutative version of the Lenard operators L n , n ≥ 1 is introduced in [14] and [8]. In this last paper a matrix Painlevé II hierarchy is studied as well, but the independent variable is left scalar. Finally we define our fully noncommutative Painlevé II hierarchy as follows PII (n) NC : where now the variable S is the diagonal matrix S := diag(s 1 , . . . , s r ) so that the anti-commutator in the right hand side is needed (also note that d dS S = I r ). In this work, first of all, we found out that the hierarchy (1.7) admits an isomonodromic Lax pair with Lax matrices that are block-matrices of dimension 2r. Furthermore, they are explicitely written in terms of the matrix valued Lenard operators defined in (1.5). The result proved in Section 4 is resumed in the following proposition. Proposition 1.1. For each fixed n there exist two polynomial matrices in λ, namely L (n) , M (n) , respectively of degree 1 and 2n, s.t. the following system is an isomonodromic Lax pair for the n−th equation of the matrix Painlevé II hierarchy (1.7).
This result can be thought as the noncommutative analogue of the well known isomonodromic Lax pair for the scalar Painlevé II hierarchy studied in [5], and resulting from self-similarity reduction of the Lax pair for the modified KdV hierarchy. Finally, we construct a solution Ψ (n) for this Lax pair (1.8), by using the solutions of the Riemann-Hilbert problems (2.1) involved in the study of the integrable operators associated to the matrix convolution operators square Ai 2 2n+1 . As by product, we obtain the relation between some solutions of the hierarchy (1.7) and the Fredholm determinants (1.4). This is indeed the final result of this work and it is proved at the end of Section 4.

Corollary 1.2.
There exists a solution W (n) of the n−th member of the matrix PII hierarchy (1.7), correspondent to the monodromy data of the Riemann-Hilbert problem (2.1), that is connected to Fredholm determinant of the n−th Airy matrix convolution operator through the following formula This result is a generalization of well known results in the scalar case. Indeed, in this case the n−th Airy kernels, defined as  [7], [6], [9]). This study turns out to be very interesting since it creates a connection between integrable systems and determinantal point processes. Indeed, the Airy kernel defines the so called Airy point process, that arises out in many areas of mathematics, such as statistichal mechanics models and random matrix theory (here some exemples of literature [19], [21], [12]). For Airy kernels (1.13) with n > 1, a generalization of this kind of results has been recently studied in [15], and in [4].
In this work, we see that the matrix Airy convolution operators squared Ai 2 2n+1 can actually be interpreted as kernels for determinantal point processes on the space of configuration {1, . . . , r} × R (under certain assumptions on the matrix C = (c j,k ) r j,k=1 ), and it would be interesting to study whether they describe phenomena in random matrix theory or statistical mechanics.
Here a more precise list of what it is done in this work.
• In Section 2 the general theory developped in [3] is applied to the operators Ai 2 2n+1 , in order to associate the Fredholm determinants (1.4) to the ones of certain integrable operators. The most important consequence of this study is indeed Theorem 2.3, that establishes a relation between Fredholm determinants (1.4) and the solutions of Riemann-Hilbert problems (2.1). Furthermore, in this section it is provided in which hypothesis these solutions exist (Theorem (2.4)), and so the relation for the Fredholm determinants found in Theorem 2.3 holds.
• In Section 3 the fully noncommutative Painlevé II hierarchy is introduced and the first equations are explicitely written.
• In the first part of Section 4, the proof of Proposition 1.1 is given and the construction of the solution Ψ (n) of the isomonodromic Lax pair (1.8) for the hierarchy (1.7) is implemented. Finally in the end of Section 4, Corollary 1.2 is proved, by using the Theorem 2.3 and the properties of the solution Ψ (n) of the isomonodromic Lax pair (1.8).

Riemann Hilbert problems associated to the matrix Airy convolution operators
In this section we are going to study the Fredholm determinants of some matrix-valued Airy convolution operator. By using the theory developped in [3] we can associate to this sequence of matrix Airy convolution operators a sequence of integrable operators with certain kernels, such that their Fredholm determinants are equal. Properties of this kind of integral kernels are studied through Riemann-Hilbert problems. As by product, this procedure allows to find the fundamental relation between Fredholm determinants of the Airy matrix convolution operators and the first asymptotic coefficient of the solutions of these Riemann-Hilbert problems, as proved in Theorem 2.3. To start with, we recall some basic fact about the scalar generalized Airy functions Ai 2n+1 . For each n ∈ N, we consider these functions Ai 2n+1 as the contour integrals where γ n ± are curves in the upper (lower) complex plane with asymptotics at ±∞ that are φ n ± := π 2 ± πn 2n + 1 , and such that γ n − = −γ n + . An exemple of these curves for n = 1 is given in Figure 1.
Definition 2.1. The n−th matrix-valued Airy function is defined as Here C = {c j,k } r j,k=1 ∈ Mat (r × r, C) and the parameters s l ∈ R, l = 1, . . . , r. With these functions we construct the matrix-valued convolution operators we are going to study in the following. Componentwises the n−th convolution operator Ai 2n+1 , looks like
• Finally, we can define the generalized matrix Airy function as where the integral is computed entry by entry.
We are actually interested in the square of the matrix Airy convolution operators defined above in (2.3). It will be indeed the Fredholm determinant of these squared operators, to be related to the fully noncommutative Painlevé II hierarchy defined in Section 2.
We are now going to define a sequence of Riemann-Hilbert problems related to the matrix Airy convolution operators. These are indeed the building blocks necessary to find the relation between Fredholm determinants of the matrix Airy convolution operators and our noncommutative Painlevé II hierarchy.
Remark 2.2. From now on, in order to simplify the notation, the dependence on s in the quantities (2.5), (2.6) will be omitted and we will use the abbreviation r (n) (λ, λ, s) = r (n) (λ).

Problem 2.1. Find a (λ−)analytic matrix valued function
it satisfies the following two conditions: − , approaching the boundary from left (+) and right (−).
• the asymptotic condition for |λ| → ∞ (2.9) Remark 2.3. In the following we are going to use the Pauli's tensorized matrices, that have the same property as the ones in the usual Pauli's algebra. In particular we denote the tensorized matrices byσ Then the standard relations hold also in this case: The following symmetry property will be useful in the next computations.
Corollary 2.2. The asymptotic coefficient appearing in the equation (2.9) have the following form

An analogue statement is true for the asymptotics coefficients of the inverse of the solution of the Riemann-Hilbert problem
Proof. We first prove the symmetry condition for the asymptotic coefficients of Ξ (n) . We start observing that the jump matrix J (n) has the following symmetrŷ just using the definition of γ n − = −γ n + . This directly implies that also the solution of the Riemann-Hilbert problem 2.1 has the same symmetry property. Thus we have that Computing the asymptotic expansion at ∞ of both sides of this equation, we have that This directly implies the two equations (2.13) for k = 2j or k = 2j − 1. Concerning the statement for the asymptotic coefficients of the inverse of Ξ (n) , namely Θ (n) , the proof follows by the fact that Θ (n) solves another Riemann-Hilbert problem, with same symmetry for the jump matrix. Indeed, consider the following problem for a function Θ (n) : • it has a jump condition for each λ ∈ γ n • it has the asymptotic condition for |λ| → ∞ The function Θ (n) with these properties is the inverse of the solution of the problem 2.1. Indeed: the functions Θ (n) Ξ (n) (λ), and Ξ (n) Θ (n) have no jumps along γ n + ∪ γ n − and they both behave like the identity matrix at ∞. Thus by the generalized Liouville theorem, they both have to coincide with the identity matrix. We then observe that the jump matrix H (n) here has the same symmetry property of J (n) , i.e.
Thus, exactly as before, even the function Θ (n) has the same property:σ We conclude then that the asymptotic coefficients of Θ (n) have the same form of the Ξ k , i.e.
We are now ready to state the fundamental result that connects the matrix Airy convolution operators to these Riemann-Hilbert problems. As far as the solutions of the Riemann-Hilbert problems 2.1 and their inverse exist, we have the following result. . Then the following identities hold Proof. The proof follows as an application to this very specific case of some general result obtained in [3]. We split the proof in two parts, one for each identity.
• In order to obtain the first identity we need essentially two results. The first one establishes the relation between Fredholm determinants of the Airy matrix convolution operators and Fredholm deteminants of certain integral kernel operators (Corollary 2.1 in [3]). In particular, we first get that the Fredholm determinants of {Ai 2n+1 } n∈N are equals to the ones of the integral operators acting on L 2 γ (n) with r (n) (λ, µ) defined as in (2.6). As by product we then have that The second result needed comes from the study of matrix integral kernels of type (2.21), through Riemann-Hilbert problems. Indeed, it allows to compute the Fredholm determinants of these integrable operators in terms of the solutions of Riemann-Hilbert problems 2.1. In particular, by appling Theorem 4.1 in [3], we have that Thus the first identity in the statement holds.
• For what concerns the second identity of the statement, we proceed by direct computation of the integral First of all, we observe that the jump matrix J (n) (λ, s) that appears in the jump condition (2.8), admits the factorization Thus we can easly compute the second factor appearing under the trace in the integral (2.24): We are now going to show that the integral above is actually just the formal residue at ∞ of a certain function. Furthermore in this particular case, due to the form of the matrix J (n) , the residue can be explicitely computed using the equation (2.25).
To start with, we consider the following function Its formal residue at ∞ can be computed as Now, this anticlockwise circle for R → ∞, can be deformed like γ (n) As by product, the formal residue of (2.26) written above, can be rewritten as the sum of two pieces, taking into account the boundary values of Θ (n) and Ξ (n) ′ along the curves γ (n) ± . We then have The second integral appearing here is actually equal to 0, since we do not have any poles at λ = 0 for the function defined in (2.26). For what concerns the first term, from the jump condition (2.8) we deduce that all along the curves γ (n) ± we have the relation Thus replacing it in the first integral above we get where in the last passages we just use the invariance of the trace by conjugation and the fact that the quantity (J (n) ) −1 (J (n) ) ′ iλσ 3 is trace free. Finally, using the asymptotic expansion at ∞ given in (2.9), we get that and that concludes the proof.
Remark 2.5. In the study of isomonodromy deformations, the quantity is associated to the isomonodromic tau function τ Ξ (n) related to the Riemann-Hilbert problem 2.1 depending on the parameters {s k } r k=1 , through the formula This notion was first introduced in [11], and then generalized for example in [2]. With Theorem 2.3 we recover for any Airy matrix convolution operator (2.3) the relation between Fredholm determinants F (n) (s 1 , . . . , s r ) and isomonodromic tau function associated to the Riemann-Hilbert problem 2.1, that was proved in Theorem 4.1 of [3] for Fredholm determinants of generic matrix convolution operators.
Finally, in order to use the formula (2.20) for the logarithmic derivative of F (n) (s 1 , . . . , s r ), we need to find out whether the solutions Ξ (n) of the Riemann-Hilbert problems 2.1 exist or not. In particular, we are going to see that under certain assumptions on the constant matrix C, the existence of Ξ (n) is assured. The following result is indeed a generalization of Theorem 5.1 in [3], for the all generalized Airy matrix convolution operators defined in (2.3). Proof. Here we consider Ai 2n+1 the scalar convolution operator acting on any f ∈ L 2 (R, C) as where inside the integral we have the scalar Airy function Ai 2n+1 defined in (2.1), without any shift and for real values of x. Now, we introduce the standard Fourier transform F and its inverse F −1 , defined as We can finally provide a complete proof of Theorem 2.4.
Proof. By applying Theorem 3.1 of [3] to the sequence of operators K (n) 2 , we have that the solutions Ξ (n) of the Riemann-Hilbert problem 2.1 exist if and only if the operator Id γ + − K (n) 2 is invertible. This is guaranteed by the non vanishing condition of the quantity det Id − K (n) 2 = det Id − Ai 2 2n+1 , that is verified if the convolution operators are such that |||Ai 2n+1 ||| < 1. Supposing that the eigenvalues of C are in the interval [−1, 1] , we are going to show that this last inequality holds for any Ai 2n+1 . Since the operators Ai 2n+1 defined in (2.3), are constructed by shifting by some component of s the Airy function, we first observe that: where Ai 0 2n+1 is the operator without any shift, namely and P s is the orthogonal projection so the sufficiency is proved.
In order to prove the other implication, we suppose that there exist λ 0 eigenvalue of C such that |λ 0 | > 1, with corresponding eigenvector v 0 ∈ C r . In this case, we will be able to construct a function f s (x) such that there exist a value s 0 for which where K Ai 2n+1 is the n−th generalized scalar Airy kernel. The corresponding kernel operator is self-adjoint and trace-class acting on L 2 ([s, ∞)). We consider its maximum eigenvalue µ(s) and the corresponding eigenfunction f s (x). Finally by taking f s ( Since λ 2 0 > 1 and µ(s) is a continuos function such that µ(s) → 1 for s → −∞ and µ(s) → 0 for s → +∞, there exist a value s 0 ∈ R for which the above equation reads as So even the necessity is proved.
Remark 2.6. As by product of the theorem above, we have that the operators Ai 2 2n+1 are bounded from above by the identity. We can actually show that any of the operators Ai 2 2n+1 is also limited from below: indeed they are all totally positive on C := {1, . . . , r} × R. The main idea to show this is to interpret Ai 2 2n+1 as a scalar function on C × C, in this way: for any couple (ξ 1 , ξ 2 ) = ((j 1 , x 1 ), (j 2 , x 2 )) ∈ C × C we have In this way the claim is proved if we prove that for any natural L, the quantity det Ai 2 2n+1 (ξ a , ξ b ) a,b≤L is positive. In order to do this, we first rewrite Ai 2 2n+1 (ξ 1 , ξ 2 ) using the product measure dµ(ξ) on C given by the product of the counting measure on {1, . . . , r} and the Lebesgue measure on R. Thus where we defined the function F 2n+1 (ξ a , ζ) = c ja,k Ai 2n+1 (x 1 + z + s ja + s k ). In this way we can determine the sign of the determinant, indeed where in the first passage we used a general property in measure theory, the Andreief equality (see here [1] for details), and in the last one we used the fact that C is hermitian.
In conclusion, by taking C an hermitian matrix with eigenvalues laying in the interval [−1, 1], any Ai 2 2n+1 is hermitian and thanks to the Theorem (2.4) and the previous remark, we can say that any Ai 2 2n+1 defines a determinantal point processes on that space of configuration C (directly by applying Theorem 3 of [18]). In particular this implies that the Fredholm determinants F (n) (s 1 , . . . , s r ) are the joint probability of the last points for some multi-process on R, (see for instance Proposition 2.9 of [13]), namely (2.45)

Matrix Painlevé II hierarchy
In this section, we are finally going to define our fully noncommutative Painlevé II hierarchy. In the following, we will consider W ( s) as a function depending on the parameters s 1 , . . . , s r with values in M at (r × r, C).
In this context we will use the standard notation for the commutator and anticommutator between two matrices: [A, ·] = A · − · A and [A, ·] + = A · + · A.
In order to define a fully noncommutative version of the PII hierarchy, we first define a matrix version of the Lenard operators. Following [8], [14]: Here I r denotes the identity matrix and d dS

(3.2)
From n ≥ 3 the "noncommutative" character of these operators appears in form of anticommutators.
Remark 3.1. In the example above and in the following we use the shorter notation d dS n W = W nS for any n ∈ N. In particular we will study the homogeneus hierarchy, setting α n = 0 for each n.
Remark 3.2. It is also possible to define a more general hierarchy, in the following way for some scalar t 1 , . . . , t n−1 . We recover the hierarchy (3.3) setting up these scalars to 0. Another matrix hierarchy was introduced in [8], but there the time variable is a scalar.

Example 3.2.
Here the first three equations of the homogeneus hierarchy (3.3).
• For n = 1 we obtain the noncommutative analogue of the homogeneus PII equation: This coincides with the homogeneus version of the fully noncommutative PII equation studied in [17], in a more general context of any noncommutative algebra with derivation.
• For n = 2 we have the 4−th order equation: (3.7) A fundamental property of matrix Lenard operators (that we are going to use in the next section in order to find the Lax pair for the hierarchy (3.3)) is given by the following formula (see [8]) .

The isomonodromic Lax pair
In this section we are finally going to find out a Lax pair for the noncommutative hierarchy (3.3), making use of the Riemann-Hilbert problems 2.1 introduced in section 2. In this way we will also be able to show the relation between some solution of the hierarchy (3.3) and the Fredholm determinant of the matrix-valued n−th Airy convolution operator. To start with, we consider a new sequence of functions, defined using the solutions of the Riemann-Hilbert problems 2.1. (4.1) It's easy to check that these functions Ψ (n) n∈N actually solve a new sequence of Riemann-Hilbert problems, with constant jump conditions. Namely, the following problems.

Problem 4.1. Find a (λ−)analytic matrix valued function
s.t. it satisfies the following two conditions: • the jump condition for each λ ∈ γ n + ∪ γ n − Ψ (n) • the asymptotic condition for |λ| → ∞ As it is standard in the theory of isomonodromic deformations, we deduce the Lax pair for the noncommutative PII hierarchy (3.3) from the Riemann-Hilbert problems with constant jumps solved by Ψ (n) . The main idea is the following: using the fact that each Ψ (n) has constant jump condition (i.e. the jump matrices K (n) do not explicitely depend on the spectral parameter λ or the deformations paramenters s i , i = 1, . . . , r), we can thus conclude that the quantities: are matrix valued polynomials in λ.
Remark 4.1. Here the inverse of Ψ (n) is simply given by Once achieved that property, using the symmetries of the Riemann-Hilbert problems 2.1, we can compute the exact form of the coefficients of these polynomials L (n) , M (n) . The final result is resumed in the proposition below.

Proposition 4.2.
There exist two polynomial matrices in λ, which we denote with L (n) and M (n) , respectively of degree 1 and 2n, such that the following system of differentail equations is satisfied: .

(4.6)
Moreover, L (n) and M (n) have the following forms and The matrix valued function L (n) is entire in λ, since it has no jumps along γ n + ∪ γ n − . Furthermore, its asymptotic behavior at infinity is given by a matrix polynomial of degree 1 in λ. Thus, by the generalized Liouville theorem, we conclude that L (n) is exactly a matrix-valued polynomial of degree 1 in λ.
In particular from the asymptotic expansion at ∞, we find an explicit form of its matrix coefficients.
where in the last two passages we used the fact that Θ (n) 1 and then the symmetry (2.13). We can then consider the second quantity defined in (4.4), namely We use the same argument as for L (n) . Indeed, also M (n) is entire in λ, since it has no jumps along γ n + ∪ γ n − . Its asymptotic behavior at infinity is given by a matrix polynomial of degree 2n in λ. We thus conclude, by the generalized Liouville theorem, that M (n) is exactly a matrix polynomial in λ of degree 2n. In particular from the asymptotic expansion at ∞ we can find an explicit form of this matrix.
(4.12) In order to obtain the remaining part of the statement, we use the following lemma.
Proof. The proof is a direct consequence of the symmetry property that the asymptotics coefficients of Ξ (n) , Θ (n) have. We start with the even case l = 2m. The coefficient of the term λ 2n−2m in the matrix M (n) is given by the following sum: where in the last sum we can find just terms like Using the symmetries (2.13) and (2.19), it's just a quick computation that shows that these terms always are a linear combination of the Pauli's matricesσ 2 ,σ 3 . So we can conclude that M where the functions A 2n−2m ( s), G 2n−2m ( s) depend on the asymptotic coefficients of Ξ (n) , Θ (n) . We work in the same way for the odd case, l = 2m − 1. The coefficient of λ 2n−2m+1 is given by the same formula where in the last sum there are just terms of the two following type In both of the cases, always replacing the symmetries (2.13) and (2.19), they result to be linear combinations of I 2r ,σ 1 . Thus we can finally conclude that Thanks to this lemma, the form of the matrix M (n) is exactly the one of the statement and the proposition is completely proved.
Remark 4.2. The system (4.6) for Ψ (n) describes the isomonodromic deformations w.r.t. the deformation parameters s i , i = 1, . . . , r, of the linear differential equation that has only one irregular singular point at ∞ of Poincaré rank r = 2n + 1, and in the special case of symmetry We can finally state that the system (4.6) is an isomonodromic Lax pair for the matrix PII hierarchy (3.3).

Proposition 4.4.
For each fixed n, the compatibility condition of the system (4.6), i.e. the equation is equivalent to the following equation

18)
Furthermore, the coefficients of the matrix M (n) are written in terms of the matrix Lenard operators in the following way Proof. We first rewrite the compatibility condition (4.17) as the following system of differential equations for the coefficients A, F, G, E: (4.20) These equations must be satisfied identically in λ. Thus, by the polinomiality of the coefficients A, F, G, E, this system is equivalent to the following one for k = 1 . . . , n. (4.21) In order to prove the statement, we are going to prove by induction over l = 2n − j that each coefficient A 2n−2k , E 2n−2k+1 , G 2n−2k , F 2n−2k+1 is given by the formulas (4.19) and that this implies that the last equation Integrating and taking the constant of integration another time equal 0 (for the same reason used above) we get The same that is given by the formula Thus for k = 1 the formulas in (4.19) gives solutions of the system (4.21). Now we proceed by induction: supposing that for l = 2n−2k+1 the coefficients E 2n−2k+1 , F 2n−2k+1 are given by the formulas (4.19), we will find that then also the coefficients for l = 2n − 2k and l = 2n − 2k − 1 have the form given by the formulas (4.19 where in the last line we used another time the property (3.8) of the matrix Lenard operators. Finally, the formula for E 2n−2k−1 directly follows from the equation above and taking the integration contant equal 0, while integrating the equation (4.21).
In the end, when we replace the formulas for G 0 , A 0 in the last equation We can then state and prove the final result of this study, that links solutions of the homogeneus matrix Painlevé II hierarchy (3.3) to Fredholm determinants of the matrix Airy convolution operators.