51 research outputs found
Universality of random matrix dynamics
We discuss the concept of width-to-spacing ratio which plays the central role
in the description of local spectral statistics of evolution operators in
multiplicative and additive stochastic processes for random matrices. We show
that the local spectral properties are highly universal and depend on a single
parameter being the width-to-spacing ratio. We discuss duality between the
kernel for Dysonian Brownian motion and the kernel for the Lyapunov matrix for
the product of Ginibre matrices.Comment: 15 pages, 3 figure
Quaternionic R transform and non-hermitian random matrices
Using the Cayley-Dickson construction we rephrase and review the
non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I.
Zahed, Nucl.Phys. B , 603 (1997)], that generalizes the free
probability calculus to asymptotically large non-hermitian random matrices. The
main object in this generalization is a quaternionic extension of the R
transform which is a generating function for planar (non-crossing) cumulants.
We demonstrate that the quaternionic R transform generates all connected
averages of all distinct powers of and its hermitian conjugate :
\langle\langle \frac{1}{N} \mbox{Tr} X^{a} X^{\dagger b} X^c \ldots
\rangle\rangle for . We show that the R transform for
gaussian elliptic laws is given by a simple linear quaternionic map
where
is the Cayley-Dickson pair of complex numbers forming a quaternion
. This map has five real parameters , ,
, and . We use the R transform to calculate the limiting
eigenvalue densities of several products of gaussian random matrices.Comment: 27 pages, 16 figure
Dynamics of Wealth Inequality
We study an agent-based model of evolution of wealth distribution in a
macro-economic system. The evolution is driven by multiplicative stochastic
fluctuations governed by the law of proportionate growth and interactions
between agents. We are mainly interested in interactions increasing wealth
inequality that is in a local implementation of the accumulated advantage
principle. Such interactions destabilise the system. They are confronted in the
model with a global regulatory mechanism which reduces wealth inequality. There
are different scenarios emerging as a net effect of these two competing
mechanisms. When the effect of the global regulation (economic interventionism)
is too weak the system is unstable and it never reaches equilibrium. When the
effect is sufficiently strong the system evolves towards a limiting stationary
distribution with a Pareto tail. In between there is a critical phase. In this
phase the system may evolve towards a steady state with a multimodal wealth
distribution. The corresponding cumulative density function has a
characteristic stairway pattern which reflects the effect of economic
stratification. The stairs represent wealth levels of economic classes
separated by wealth gaps. As we show, the pattern is typical for macro-economic
systems with a limited economic freedom. One can find such a multimodal pattern
in empirical data, for instance, in the highest percentile of wealth
distribution for the population in urban areas of China.Comment: 17 pages, 8 figures (references added, some material moved to
appendix
Universal distribution of Lyapunov exponents for products of Ginibre matrices
Starting from exact analytical results on singular values and complex
eigenvalues of products of independent Gaussian complex random
matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an
infinite product. We show that for a large number of product matrices the
distribution of each Lyapunov exponent is normal and compute its -dependent
variance as well as corrections in a expansion. Originally Lyapunov
exponents are defined for singular values of the product matrix that represents
a linear time evolution. Surprisingly a similar construction for the moduli of
the complex eigenvalues yields the very same exponents and normal distributions
to leading order. We discuss a general mechanism for matrices why
the singular values and the radii of complex eigenvalues collapse onto the same
value in the large- limit. Thereby we rederive Newman's triangular law which
has a simple interpretation as the radial density of complex eigenvalues in the
circular law and study the commutativity of the two limits and
on the global and the local scale. As a mathematical byproduct we
show that a particular asymptotic expansion of a Meijer G-function with large
index leads to a Gaussian.Comment: 36 pages, 6 figure
From 4D Reduced SYM Integrals to Branched-Polymers
We derive analytically one-loop corrections to the effective Polyakov-line
operator in the branched-polymer approximation of the reduced four-dimensional
supersymmetric Yang-Mills integrals.Comment: to be published in Acta Physica Polonic
Semiclassical Geometry of 4D Reduced Supersymmetric Yang-Mills Integrals
We investigate semiclassical properties of space-time geometry of the low
energy limit of reduced four dimensional supersymmetric Yang-Mills integrals
using Monte-Carlo simulations. The limit is obtained by an one-loop
approximation of the original Yang-Mills integrals leading to an effective
model of branched polymers. We numerically determine the behaviour of the
gyration radius, the two-point correlation function and the Polyakov-line
operator in the effective model and discuss the results in the context of the
large-distance behaviour of the original matrix model.Comment: 14 pages, 5 figures, corrected version v
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