949 research outputs found

### Pfaffian Expressions for Random Matrix Correlation Functions

It is well known that Pfaffian formulas for eigenvalue correlations are
useful in the analysis of real and quaternion random matrices. Moreover the
parametric correlations in the crossover to complex random matrices are
evaluated in the forms of Pfaffians. In this article, we review the
formulations and applications of Pfaffian formulas. For that purpose, we first
present the general Pfaffian expressions in terms of the corresponding skew
orthogonal polynomials. Then we clarify the relation to Eynard and Mehta's
determinant formula for hermitian matrix models and explain how the evaluation
is simplified in the cases related to the classical orthogonal polynomials.
Applications of Pfaffian formulas to random matrix theory and other fields are
also mentioned.Comment: 28 page

### On the correlation function of the characteristic polynomials of the hermitian Wigner ensemble

We consider the asymptotics of the correlation functions of the
characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$.
We show that for the correlation function of any even order the asymptotic
coincides with this for the GUE up to a factor, depending only on the forth
moment of the common probability law $Q$ of entries $\Im W_{jk}$, $\Re W_{jk}$,
i.e. that the higher moments of $Q$ do not contribute to the above limit.Comment: 20

### Level-Spacing Distributions and the Bessel Kernel

The level spacing distributions which arise when one rescales the Laguerre or
Jacobi ensembles of hermitian matrices is studied. These distributions are
expressible in terms of a Fredholm determinant of an integral operator whose
kernel is expressible in terms of Bessel functions of order $\alpha$. We derive
a system of partial differential equations associated with the logarithmic
derivative of this Fredholm determinant when the underlying domain is a union
of intervals. In the case of a single interval this Fredholm determinant is a
Painleve tau function.Comment: 18 pages, resubmitted to make postscript compatible, no changes to
manuscript conten

### Dyson's Brownian Motion and Universal Dynamics of Quantum Systems

We establish a correspondence between the evolution of the distribution of
eigenvalues of a $N\times N$ matrix subject to a random Gaussian perturbing
matrix, and a Fokker-Planck equation postulated by Dyson. Within this model, we
prove the equivalence conjectured by Altshuler et al between the space-time
correlations of the Sutherland-Calogero-Moser system in the thermodynamic limit
and a set of two-variable correlations for disordered quantum systems
calculated by them. Multiple variable correlation functions are, however, shown
to be inequivalent for the two cases.Comment: 10 pages, revte

### A method to calculate correlation functions for $\beta=1$ random matrices of odd size

The calculation of correlation functions for $\beta=1$ random matrix
ensembles, which can be carried out using Pfaffians, has the peculiar feature
of requiring a separate calculation depending on the parity of the matrix size
N. This same complication is present in the calculation of the correlations for
the Ginibre Orthogonal Ensemble of real Gaussian matrices. In fact the methods
used to compute the $\beta=1$, N odd, correlations break down in the case of N
odd real Ginibre matrices, necessitating a new approach to both problems. The
new approach taken in this work is to deduce the $\beta=1$, N odd correlations
as limiting cases of their N even counterparts, when one of the particles is
removed towards infinity. This method is shown to yield the correlations for N
odd real Gaussian matrices.Comment: 20 pages, corrected typo

### Moments of a single entry of circular orthogonal ensembles and Weingarten calculus

Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$
from a circular orthogonal ensemble. In this paper, we study moments of a
single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit
values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading
and subleading terms in the asymptotic expansion with respect to a large matrix
size $N$. Our technique is to apply the Weingarten calculus for a
Haar-distributed unitary matrix.Comment: 17 page

### The factorization method for systems with a complex action -a test in Random Matrix Theory for finite density QCD-

Monte Carlo simulations of systems with a complex action are known to be
extremely difficult. A new approach to this problem based on a factorization
property of distribution functions of observables has been proposed recently.
The method can be applied to any system with a complex action, and it
eliminates the so-called overlap problem completely. We test the new approach
in a Random Matrix Theory for finite density QCD, where we are able to
reproduce the exact results for the quark number density. The achieved system
size is large enough to extract the thermodynamic limit. Our results provide a
clear understanding of how the expected first order phase transition is induced
by the imaginary part of the action.Comment: 27 pages, 25 figure

### Amplification of SOX4 promotes PI3K/Akt signaling in human breast cancer

Purpose: The PI3K/Akt signaling axis contributes to the dysregulation of many dominant features in breast cancer including cell proliferation, survival, metabolism, motility, and genomic instability. While multiple studies have demonstrated that basal-like or triple-negative breast tumors have uniformly high PI3K/Akt activity, genomic alterations that mediate dysregulation of this pathway in this subset of highly aggressive breast tumors remain to be determined. Methods: In this study, we present an integrated genomic analysis based on the use of a PI3K gene expression signature as a framework to analyze orthogonal genomic data from human breast tumors, including RNA expression, DNA copy number alterations, and protein expression. In combination with data from a genome-wide RNA-mediated interference screen in human breast cancer cell lines, we identified essential genetic drivers of PI3K/Akt signaling. Results: Our in silico analyses identified SOX4 amplification as a novel modulator of PI3K/Akt signaling in breast cancers and in vitro studies confirmed its role in regulating Akt phosphorylation. Conclusions: Taken together, these data establish a role for SOX4-mediated PI3K/Akt signaling in breast cancer and suggest that SOX4 may represent a novel therapeutic target and/or biomarker for current PI3K family therapies

### The Large-$N$ Limit of the Two-Hermitian-matrix model by the hidden BRST method

This paper discusses the large N limit of the two-Hermitian-matrix model in
zero dimensions, using the hidden BRST method. A system of integral equations
previously found is solved, showing that it contained the exact solution of the
model in leading order of large $N$.Comment: 19 pages, Latex,CERN--TH-6531/9

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