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Register a new userGravitational lensing by eigenvalue distributions of random matrix models
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Added by
Luis Martinez Alonso
Publication date
2018
fields
Physics
and research's language is
English
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We propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions in gravitational lensing. The corresponding lens equations reduce to algebraic equations in the complex plane which can be treated analytically. We prove that these models can be applied to describe lensing by systems of edge-on galaxies. We illustrate our analysis with the Gaussian and the quartic unitary matrix ensembles.
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We introduce a simple yet powerful calculational tool useful in calculating averages of ratios and products of characteristic polynomials. The method is based on Dyson Brownian motion and Grassmann integration formula for determinants. It is intended as an alternative to other RMT techniques applicable to general gaussian measures. Resulting formulas are exact for finite matrix size N and form integral representations convenient for large N asymptotics. Quantities obtained by the method can be interpreted as averages over matrix models with an external source. We provide several explicit and novel calculations showing a range of applications.
Let $mathbf{W}_1$ and $mathbf{W}_2$ be independent $ntimes n$ complex central Wishart matrices with $m_1$ and $m_2$ degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices $(mathbf{W}_1+mathbf{W}_2)^{-1}mathbf{W}_1$, which are analogous to those of F matrices ${bf W}_1 {bf W}_2^{-1}$ and those of the Jacobi unitary ensemble (JUE). Defining $alpha_1=m_1-n$ and $alpha_2=m_2-n$, we derive new exact distribution formulas in terms of $(alpha_1+alpha_2)$-dimensional matrix determinants, with elements involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-$n$ analysis with $alpha_1$ and $alpha_2$ fixed (i.e., under the so-called hard-edge scaling limit); the analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as $n to infty$ in terms of $alpha_1$- and $alpha_2$-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-$n$ corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painleve differential equations, or hypergeometric functions of matrix arguments.
The purpose of the present paper is to derive a partial differential equation (PDE) for the single-time single-point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly non-linear parabolic-transport equation, which depends on two conditional statistical numerics of important physical significance. The PDF PDE is a general form of the classical Reynolds mean flow equation, and is a precise formulation of the PDF transport equation. The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed, though the example is seemingly artificial, but it demonstrates the PDF method based on the new PDF PDE.
Using the methods originally developed for Random Matrix Theory we derive an exact mathematical formula for number variance (introduced in [4]) describing a rigidity of particle ensembles with power-law repulsion. The resulting relation is consequently compared with the relevant statistics of the single-vehicle data measured on the Dutch freeway A9. The detected value of an inverse temperature, which can be identified as a coefficient of a mental strain of car drivers, is then discussed in detail with the respect to the traffic density and flow.
We give a generalization of the random matrix ensembles, including all lassical ensembles. Then we derive the joint density function of the generalized ensemble by one simple formula, which give a direct and unified way to compute the density functions for all classical ensembles and various kinds of new ensembles. An integration formula associated with the generalized ensemble is also given. We also give a classification scheme of the generalized ensembles, which will include all classical ensembles and some new ensembles which were not considered before.
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