Generalized Christoffel–Darboux formula for skew-orthogonal polynomials and random matrix theory

Abstract

We obtain a generalized Christoffel-Darboux (GCD) formula for skew-orthogonal polynomials. Using this, we present an alternative derivation of the level density and two-point function for Gaussian orthogonal ensembles and Gaussian symplectic ensembles of random matrices.PACS numbers: 02.30. Gp, 05.45.Mt Random matrices have found applications in different branches of physics mainly due to the 'universality' in their correlation function under certain scaling limits. In this context, although unitary ensembles of random matrices, corresponding to systems with broken time-reversal symmetry (for example, a mesoscopic conductor in the presence of a magnetic field) have been extensively studied [1,2], much less is known about Orthogonal Ensembles (OE) and Symplectic Ensembles (SE) of random matrices.In his phenomenal paper [3] Dyson stressed the importance of a good understanding of the skew-orthogonal polynomials (SOP) to study the OE and SE of random matrices. Some progress has been made in this regard [1,4,5] to develop the theory of SOP. Alternative approaches were taken by various authors [6,7,8,9,10,11,12,13] to study these ensembles. For example, Deift and Gioev [14,15] have recently used the Widom's representation [16] to prove Universality for a wide class of polynomial potentials of the OE and SE of random matrices. In all these work, the authors have used the well known properties of orthogonal polynomials to study these ensembles. Here we use the more elegant SOP, evolving naturally from OE and SE of random matrices.In this paper we obtain recursion relations for SOP. Using this we derive the GCD formula. We use them to obtain the level density and two-point correlation function for Gaussian orthogonal ensembles and Gaussian symplectic ensembles of random matrices.We consider ensembles of 2N dimensional matrices H with probability distributionwhere the parameter β = 1 and 4 corresponds to H real symmetric or quaternion real self dual.(Note that we have considered the OE of even dimension. The odd dimension can be easily generalized.) dH is the standard Haar measure. Z βN is the so called 'partition function', and is proportional to the product of skew-normalization constants [1]:Here n F is called the 'Fermi level' by analogy with a system of fermions. In our case, it takes the value n F = 2N for β = 1 and 4 respectively. It is assumed that most of the physical quantities (like level density and correlation functions) are related to properties of g (β)

Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential and universality in the matrix model

“…Here, we note, that for a given x, θ varies with m. We have chosen θ ≡ θ 2m+2 such that 8) and so on for φ (4) 2m±k (x), ψ…”

Analysis of Terahertz Oscillator Using Negative Differential Resistance Dual-Channel Transistor and Integrated Antenna

“…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 [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. From mathematical point of view, the study of 'universality' in the energy level correlations of random matrices requires a good understanding of the asymptotic behaviour of certain families of polynomials.…”

“…The rich literature available on orthogonal polynomials [6][7][8][9][10][11][12][13][14][15][16][17][18] and bi-orthogonal polynomials [19][20][21][22][23] has contributed a lot in our understanding of the unitary ensembles. Our aim is to develop the theory of skew-orthogonal polynomials [3][4][5][24][25][26][27] so that we can have further insight into the one-matrix model for orthogonal and symplectic ensembles of random matrices.…”

Skew-orthogonal polynomials, differential systems and random matrix theory

“…[5], where we used a matrix of size 2N + 2 to prove the 'Universality' in the Gaussian case. Also the definition of R (1) and P (1) differs by a factor of 2.…”