# Skew-orthogonal polynomials, differential systems and random matrix theory

###### Abstract

We study skew-orthogonal polynomials with respect to the weight function , with , , . A finite subsequence of such skew-orthogonal polynomials arising in the study of Orthogonal and Symplectic ensembles of random matrices, satisfy a system of differential-difference-deformation equation. The vectors formed by such subsequence has the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions.

###### pacs:

02.30.Gp, 05.45.Mt## 1 Introduction

### 1.1 Random matrices

The concept of ‘Universality’ in random matrix theory and its various applications in real physical systems have attracted both mathematicians and physicists in the last few decades guhr ; beenakker ; ghosh ; ghoshpandey ; ghosh3 ; deift1 ; deift2 ; deeift3 ; baik ; bleher ; deift4 ; deift5 ; deift6 ; deift7 . From mathematical point of view, the study of ‘Universality’ in the energy level correlations of random matrices require a good understanding of the asymptotic behavior of certain families of polynomials. For example Unitary ensembles of random matrices require an understanding of the large behavior of orthogonal (the one-matrix model) and bi-orthogonal (the two matrix model) polynomials, while Orthogonal and Symplectic ensembles require that of the skew-orthogonal polynomials. The rich literature available on orthogonal polynomials szego ; deift1 ; deift2 ; deeift3 ; baik ; bleher ; deift4 ; deift5 ; deift6 ; deift7 ; eynard1 ; eynard11 ; plancherel and bi-orthogonal polynomials eynard2 ; eynard3 ; eynard4 ; Kap ; KM have contributed a lot in our understanding of the Unitary ensembles. Our aim is to develop the theory of skew-orthogonal polynomials ghosh ; ghoshpandey ; ghosh3 ; dyson1 ; mehta ; mehta1 ; eynard so that we can have further insight into the one-matrix model for Orthogonal and Symplectic ensembles of random matrices.

In this direction, previous experience with orthogonal and bi-orthogonal polynomials arising in the two-matrix model makes us believe that perhaps the most logical and rigorous way to study the asymptotic properties of these skew-orthogonal polynomials is to do the following: (1). First, we must look for a finite subsequence of skew-orthonormal vectors which will satisfy a mutually compatible system of differential-difference-deformation equation. This requires one to fix the ‘size’ or rank of this finite sub-sequence. The Generalized Christoffel-Darboux formula helps in this regard as it gives an indication of the number of terms near the so-called ‘Fermi-level’ which are actually required to study this system. (2). The next step is to look for a mutually compatible system of differential-difference-deformation equation satisfied by the finite subsequence of these vectors and hence find the so-called ‘fundamental solution’. (3). Finally, to formulate a quaternion matrix Riemann-Hilbert problem and by applying the steepest descent method, obtain the Plancherel-Rotach-type plancherel formula for skew-orthogonal polynomials.

Reference ghosh3 achieved the first goal. In this paper, we try to address the second issue of the existence of the ‘fundamental solution’ of a system of differential-difference-deformation equation. One of the by-products is that we observe certain duality property between the skew-orthonormal vectors of Orthogonal and Symplectic ensembles. (In fact, this justifies further the use of skew-orthogonal polynomials to ordinary orthogonal polynomials in studying these ensembles stoj1 ; stoj1a ; stoj2 ; deift1 ; deift2 ; nagao ; tracy1 ; tracy2 ; tracy3 ; adler ; widom .) The third part, which is also the most crucial one, is not yet fully understood.

We shall consider an ensemble of dimensional matrices with probability distribution

(1.1) |

where the parameter and corresponds to the joint probability density for Orthogonal and Symplectic ensembles of random matrices. The “potential” is a polynomial of degree with positive leading coefficient:

(1.2) |

where is called the deformation parameter. is the so called ‘partition function’:

(1.3) |

where is a set of all real symmetric () and quaternion real self dual () matrices. is the standard Haar measure. is the skew-normalization constant for polynomials related to Orthogonal and Symplectic ensembles dyson1 .

Remark on notation. Before entering the details of calculation, we must mention that throughout this paper, we have followed to a great extent notations used in eynard ; eynard1 ; eynard2 ; eynard3 ; eynard4 . Apart from having the advantage of using a well established and compact notation, this strategy will also highlight the striking similarity (as well as the difference) between skew-orthogonal polynomials and their bi-orthogonal counterpart.

### 1.2 Skew-Orthogonal Polynomials: relevance in Orthogonal and Symplectic ensembles

Definition:

For Orthogonal and Symplectic ensembles of random matrices, we define semi-infinite vectors:

(1.4) | |||

(1.5) |

where

(1.6) |

Each entry in these semi-infinite vectors is a matrix:

(1.7) |

where

(1.8) |

is the weighted skew-normalized polynomial (often called the quasi-polynomial) and

(1.9) |

is monic skew-orthogonal polynomial of order .

(1.10) |

is semi-infinite anti-symmetric block-diagonal matrix with and

(1.11) |

is the step function. These vectors satisfy skew-orthonormality relation:

(1.12) |

Remark: This definition differs from that of mehta ; ghosh ; ghoshpandey for by a factor of , which is incorporated in the normalization constant.

These skew-orthonormal vectors form the necessary constituents of the kernel functions required to study different statistical properties of Orthogonal ensembles and Symplectic ensembles of random matrices. For example, the two point correlation function can be written in terms of the kernel function mehta ; mehta1 :

(1.13) |

where is the kronecker delta and

(1.14) |

### 1.3 The Generalised Christoffel Darboux sum

Here, we present a summary of the main results of Ref.ghosh3 , where we studied the kernel function . We expand , and in terms of (and hence introduce the semi-infinite matrices , and respectively):

(1.18) | |||||

(1.19) | |||||

(1.20) |

where Eq.(1.20) is obtained by multiplying the above expansion by and integrating by parts. Here

(1.21) |

The matrices , and are quaternion matrices. For a nice introduction to the subject, the reader is referred to the book by Prof. M. L. Mehta mehta1 .

Defining the quaternion-matrix

(1.22) |

the Generalized Christoffel Darboux formula for the Symplectic ensembles () can be written as:

(1.23) |

For Orthogonal ensembles () the Generalized Christoffel Darboux is given by:

(1.24) |

The Generalized Christoffel-Darboux matrix in terms of the elements of the quaternion matrix takes the form

(1.25) |

Here, we must point out that there is a small difference with the notation used in the second part of Ref.ghosh3 , where we used a matrix of size to prove the ‘Universality’ in the Gaussian case. Also the definition of and differs by a factor of .

### 1.4 Difference-differential-deformation equations

A quick glance at Eq.(1.25) reveals that the relevant vectors contributing to the correlation function are . This prompts us to define a finite subsequence (or window: ) of skew-orthogonal vectors of size (in the quaternionic sense), and :

(1.26) | |||

(1.27) |

such that

(1.28) | |||

(1.29) |

The rank of the window is equal to the degree of the potential in the quaternionic space and twice that of the degree of in the real space.

The recursion relations connecting the finite subsequence (or window) with the upper or lower one is through the ladder operator

(1.30) |

(1.31) |

and

(1.32) |

(1.33) |

The vectors and also satisfy a system of ODEs:

(1.34) |

Under an infinitesimal change of the deformation parameter (the coefficients of the polynomial potential ), these vectors satisfy a system of PDE’s given by

(1.35) |

Thus the finite subsequence of skew-orthogonal vectors satisfy a system of difference-differential-deformation equation.

Remark. The two pairs of matrices and (similarly ( and , and (similarly ( and and and (similarly and ) are dual in the sense that they remain invariant under an interchange of and (and similarly for and ).

### 1.5 Compatibility

The existence of recursion relations, differential equations and deformation equations for vectors arising in the Orthogonal ensembles and Symplectic ensembles of random matrices can be viewed as just a projection of the semi-infinite functions and onto the finite “window” and respectively. However, we may also consider these equations as defining an overdetermined system of finite difference-differential-deformation equations of the vector functions and see that these systems are compatible. This leads to the existence of a fundamental matrix solution, denoted by and where all the column vectors satisfy the above difference-differential-deformation equations simultaneously.

The compatibility of the deformation and difference equation with the differential equation imply that the generalized monodromy of the operator and is invariant under deformations and shifts in .

Outline of the article

-Section 2 deals with the properties of different finite band matrices related to skew-orthogonal polynomials.

-In section 3, we study the system of PDE’s arising from the infinitesimal change of the deformation parameter .

-We derive the difference relations satisfied by the finite subsequence of skew-orthogonal vectors in section 4.

-In section 5, we derive the folding function which is used to project any given vector (or ) onto its finite subsequence or window.

-In section 6, we obtain the folded deformation matrix, using the results of section 5.

- The differential equation for skew-orthonormal vectors, using the results obtained in Section 6, is derived in section 7.

-In section 8, we discuss the existence of the Cauchy-like transforms of the skew-orthogonal vectors of order as the other solutions to the differential-deformation-difference equations, for fairly large .

-We prove compatibility conditions for these difference-deformation-differential system in section 9.

## 2 Recursion Relations and finite band matrices.

In this section, we will study in detail the different recursion relations satisfied by the skew-orthonormal vectors. Here, we must note that unlike the orthogonal polynomials, skew-orthonormal vectors do not satisfy a three-term recursion relation and hence do not give birth to the tri-diagonal Jacobi matrix.

(2.1) | |||

(2.2) | |||

(2.3) |

where and are semi-infinite column vectors defined in Eqs.(1.4,1.5). This is equivalent to saying

(2.4) | |||||

(2.5) |

These finite band matrices satisfy the following commutation relations:

(2.6) |

Here, each entry is a quaternion of the form:

(2.7) |

Using and for , and replacing by for , we get

(2.8) |

where dual of a matrix is defined as mehta1 :

(2.9) |

Remark. The condition of anti-self duality imposes a much lighter restriction on a diagonal quaternion than antisymmetry condition on a diagonal matrix element. For example, it leaves the off-diagonal entry of the diagonal quaternion arbitrary. This is the reason why the odd skew-orthogonal polynomials are arbitrary upto the addition of a lower even polynomial.

However starting with and using (2.6), we get

(2.10) |

As pointed out in the beginning of the section, this essentially means that like the orthogonal polynomials, we do not have a tri-diagonal Jacobi-matrix for skew-orthogonal polynomials. It is this relation that causes the Generalized Christoffel-Darboux sum to have a local behavior (i.e. the rank has dependence on the measure or weight function). In a sense, this is a major setback in our hope of defining a matrix Riemann-Hilbert problem for skew-orthogonal vectors, similar to that of the orthogonal polynomials. Also from the dependence of and on (the degree of the polynomial ), we can conclude that the ‘size’ of our q-matrix Riemann-Hilbert problem will depend on .

From ghosh3 , we also have

(2.11) |

where ‘lower’ and ‘’ denotes a strictly lower triangular matrix and a lower triangular matrix with the principal diagonal respectively.

Having obtained the recursion relations for the skew-orthonormal vectors, we introduce a convenient notation to express band-matrices and :

(2.12) |

and

(2.13) |

where

(2.14) |

Here, we have suppressed the -dependence of . This notation will be useful in deriving the folding function.

More explicitly, with this notation, we can write

(2.15) |

and

(2.16) |

We note that for even , the quaternions of the outermost band are such that

(2.17) |

From (2.6), we get the following quadratic relation between the coefficients

(2.18) |

This is the compatibility relation for the difference-differential system and will be used in Section IX.

Remark. Here, one can notice the basic difference between the properties of bi-orthogonal polynomials and their skew-orthogonal counterparts. In the skew-orthogonal vectors, the semi-infinite matrices are symmetric around the principal diagonal. This is not true for the bi-orthogonal polynomials where the matrices have an asymmetry.

## 3 The Deformation Matrix

In this section, we will consider infinitesimal deformation corresponding to changes in , the coefficient of the potential . Using the definition of the semi-infinite matrices and (and , ), the deformation matrix can be defined as:

(3.1) | |||||

(3.2) |

The matrix is anti-self dual, i.e.:

(3.3) |

Moreover, they satisfy the following relations with and (we drop the superscript from the matrix , and for simplicity):

(3.4) |

Explicitly, can be written as

(3.5) |

where , denote the upper and lower triangular quaternion matrix while is the diagonal quaternion.

Proof:

Differentiating Eq.(1.12) w.r.t. and using Eqs. (3.1) and (3.2), we get Eq.(3.3). Eq.(3.4) follows by interchanging the operators and with . Finally to prove Eq.(3.5), we start with the normalized quasi-polynomial

(3.6) |

where is defined in Eq.(1.2) and

(3.7) |

is the monic skew-orthogonal polynomial, defined in Eq.(1.9). To save cluttering, we have suppressed the dependence of . We have used . Differentiating with respect to the deformation parameter, we get

(3.8) |

Also differentiating the skew-normalization condition, we get

(3.9) | |||||

The diagonal elements have the form:

(3.10) | |||||

Thus, we get

(3.11) |

Remark. Here the operation means that the dual is taken first and then the lower triangular part collected. Also, due to the arbitrariness in the definition of an anti-self dual matrix, we may choose the lower off-diagonal element of the diagonal quaternions in zero. With this choice, we can remove the arbitrariness in the definition of the odd skew-orthonormal polynomials.