24,593 research outputs found

### 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

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