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

We derive bulk asymptotics of skew-orthogonal polynomials (sop) \pi^{\bt}_{m}, $\beta=1$, 4, defined w.r.t. the weight $\exp(-2NV(x))$, $V (x)=gx^4/4+tx^2/2$, $g>0$ and $t<0$. We assume that as $m,N \to\infty$ there exists an $\epsilon > 0$, such that $\epsilon\leq (m/N)\leq \lambda_{\rm cr}-\epsilon$, where $\lambda_{\rm cr}$ is the critical value which separates sop with two cuts from those with one cut. Simultaneously we derive asymptotics for the recursive coefficients of skew-orthogonal polynomials. The proof is based on obtaining a finite term recursion relation between sop and orthogonal polynomials (op) and using asymptotic results of op derived in \cite{bleher}. Finally, we apply these asymptotic results of sop and their recursion coefficients in the generalized Christoffel-Darboux formula (GCD) \cite{ghosh3} to obtain level densities and sine-kernels in the bulk of the spectrum for orthogonal and symplectic ensembles of random matrices.Comment: 6 page

Matrices coupled in a chain. I. Eigenvalue correlations

The general correlation function for the eigenvalues of $p$ complex hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.Comment: ftex eynmeh.tex, 2 files, 8 pages Submitted to: J. Phys.

Zeros of some bi-orthogonal polynomials

Ercolani and McLaughlin have recently shown that the zeros of the bi-orthogonal polynomials with the weight $w(x,y)=\exp[-(V_1(x)+V_2(y)+2cxy)/2]$, relevant to a model of two coupled hermitian matrices, are real and simple. We show that their argument applies to the more general case of the weight $(w_1*w_2*...*w_j)(x,y)$, a convolution of several weights of the same form. This general case is relevant to a model of several hermitian matrices coupled in a chain. Their argument also works for the weight $W(x,y)=e^{-x-y}/(x+y)$, $0\le x,y<\infty$, and for a convolution of several such weights.Comment: tex mehta.tex, 1 file, 9 pages [SPhT-T01/086], submitted to J. Phys.

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Moments of the characteristic polynomial in the three ensembles of random matrices

Moments of the characteristic polynomial of a random matrix taken from any of the three ensembles, orthogonal, unitary or symplectic, are given either as a determinant or a pfaffian or as a sum of determinants. For gaussian ensembles comparing the two expressions of the same moment one gets two remarkable identities, one between an $n\times n$ determinant and an $m\times m$ determinant and another between the pfaffian of a $2n\times 2n$ anti-symmetric matrix and a sum of $m\times m$ determinants.Comment: tex, 1 file, 15 pages [SPhT-T01/016], published J. Phys. A: Math. Gen. 34 (2001) 1-1

Probability density of determinants of random matrices

In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

Finite-difference distributions for the Ginibre ensemble

The Ginibre ensemble of complex random matrices is studied. The complex valued random variable of second difference of complex energy levels is defined. For the N=3 dimensional ensemble are calculated distributions of second difference, of real and imaginary parts of second difference, as well as of its radius and of its argument (angle). For the generic N-dimensional Ginibre ensemble an exact analytical formula for second difference's distribution is derived. The comparison with real valued random variable of second difference of adjacent real valued energy levels for Gaussian orthogonal, unitary, and symplectic, ensemble of random matrices as well as for Poisson ensemble is provided.Comment: 8 pages, a number of small changes in the tex

Accuracy and range of validity of the Wigner surmise for mixed symmetry classes in random matrix theory

Schierenberg et al. [Phys. Rev. E 85, 061130 (2012)] recently applied the Wigner surmise, i.e., substitution of \infty \times \infty matrices by their 2 \times 2 counterparts for the computation of level spacing distributions, to random matrix ensembles in transition between two universality classes. I examine the accuracy and the range of validity of the surmise for the crossover between the Gaussian orthogonal and unitary ensembles by contrasting them with the large-N results that I evaluated using the Nystrom-type method for the Fredholm determinant. The surmised expression at the best-fitting parameter provides a good approximation for 0 \lesssim s \lesssim 2, i.e., the validity range of the original surmise.Comment: 3 pages in REVTeX, 10 figures. (v2) Title changed, version to appear in Phys. Rev.

The correspondence between Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches in random matrix theory: the Gaussian case

Two approaches (TW and ASvM) to derivation of integrable differential equations for random matrix probabilities are compared. Both methods are rewritten in such a form that simple and explicit relations between all TW dependent variables and $\tau$-functions of ASvM are found, for the example of finite size Gaussian matrices. Orthogonal function systems and Toda lattice are seen as the core structure of both approaches and their relationship.Comment: 20 pages, submitted to Journal of Mathematical Physic

The Local Semicircle Law for Random Matrices with a Fourfold Symmetry

We consider real symmetric and complex Hermitian random matrices with the additional symmetry $h_{xy}=h_{N-x,N-y}$. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble (GOE). It also occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.Comment: 20 pages, to appear in J. Math. Phy