62 research outputs found
Buses of Cuernavaca - an agent-based model for universal random matrix behavior minimizing mutual information
The public transportation system of Cuernavaca, Mexico, exhibits random
matrix theory statistics [1]. In particular, the fluctuation of times between
the arrival of buses on a given bus stop, follows the Wigner surmise for the
Gaussian Unitary Ensemble. To model this, we propose an agent-based approach in
which each bus driver tries to optimize his arrival time to the next stop with
respect to an estimated arrival time of his predecessor. We choose a particular
form of the associated utility function and recover the appropriate
distribution in numerical experiments for a certain value of the only parameter
of the model. We then investigate whether this value of the parameter is
otherwise distinguished within an information theoretic approach and give
numerical evidence that indeed it is associated with a minimum of averaged
pairwise mutual information.Comment: numerical analysis extende
Universal shocks in the Wishart random-matrix ensemble - a sequel
We study the diffusion of complex Wishart matrices and derive a partial
differential equation governing the behavior of the associated averaged
characteristic polynomial. In the limit of large size matrices, the inverse
Cole-Hopf transform of this polynomial obeys a nonlinear partial differential
equation whose solutions exhibit shocks at the evolving edges of the eigenvalue
spectrum. In a particular scenario one of those shocks hits the origin that
plays the role of an impassable wall. To investigate the universal behavior in
the vicinity of this wall, a critical point, we derive an integral
representation for the averaged characteristic polynomial and study its
asymptotic behavior. The result is a Bessoid function.Comment: 7 pages, 2 figure
Universal shocks in the Wishart random matrix ensemble - 1
We show that the derivative of the logarithm of the average characteristic
polynomial of a diffusing Wishart matrix obeys an exact partial differential
equation valid for an arbitrary value of N, the size of the matrix. In the
large N limit, this equation generalizes the simple Burgers equation that has
been obtained earlier for Hermitian or unitary matrices. The solution through
the method of characteristics presents singularities that we relate to the
precursors of shock formation in fluid dynamical equations. The 1/N corrections
may be viewed as viscous corrections, with the role of the viscosity being
played by the inverse of the doubled dimension of the matrix. These corrections
are studied through a scaling analysis in the vicinity of the shocks, and one
recovers in a simple way the universal Bessel oscillations (so-called hard edge
singularities) familiar in random matrix theory.Comment: 9 page
Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD
We link the spontaneous breakdown of chiral symmetry in Euclidean QCD to the
collision of spectral shock waves in the vicinity of zero eigenvalue of Dirac
operator. The mechanism, originating from complex Burger's-like equation for
viscid, pressureless, one-dimensional flow of eigenvalues, is similar to
recently observed weak-strong coupling phase transition in large
Yang-Mills theory. The spectral viscosity is proportional to the inverse of the
size of the random matrix that replaces the Dirac operator in the universal
(ergodic) regime. We obtain the exact scaling function and critical exponents
of the chiral phase transition for the averaged characteristic polynomial for
QCD. We reinterpret our results in terms of known properties of
chiral random matrix models and lattice data.Comment: 12 page
Full Dysonian dynamics of the complex Ginibre ensemble
We find stochastic equations governing eigenvalues and eigenvectors of a
dynamical complex Ginibre ensemble reaffirming the intertwined role played
between both sets of matrix degrees of freedom. We solve the accompanying
Smoluchowski-Fokker-Planck equation valid for any initial matrix. We derive
evolution equations for the averaged extended characteristic polynomial and for
a class of -point eigenvalue correlation functions. From the latter we
obtain a novel formula for the eigenvector correlation function which we
inspect for Ginibre and spiric initial conditions and obtain macro- and
microscopic limiting laws.Comment: minor typos corrected, some references update
Diffusion in the space of complex Hermitian matrices - microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial
We show that the averaged characteristic polynomial and the averaged inverse
characteristic polynomial, associated with Hermitian matrices whose elements
perform a random walk in the space of complex numbers, satisfy certain partial
differential, diffusion-like, equations. These equations are valid for matrices
of arbitrary size. Their solutions can be given an integral representation that
allows for a simple study of their asymptotic behaviors for a broad range of
initial conditions.Comment: 26 pages, 4 figure
Hydrodynamics of the Dirac spectrum
We discuss a hydrodynamical description of the eigenvalues of the Dirac spectrum in even dimensions in the vacuum and in the large N (volume) limit. The linearized hydrodynamics supports sound waves. The hydrodynamical relaxation of the eigenvalues is captured by a hydrodynamical (tunneling) minimum configuration which follows from a pertinent form of Euler equation. The relaxation from a phase of unbroken chiral symmetry to a phase of broken chiral symmetry occurs over a time set by the speed of sound
Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
Following our recent letter, we study in detail an entry-wise diffusion of
non-hermitian complex matrices. We obtain an exact partial differential
equation (valid for any matrix size and arbitrary initial conditions) for
evolution of the averaged extended characteristic polynomial. The logarithm of
this polynomial has an interpretation of a potential which generates a Burgers
dynamics in quaternionic space. The dynamics of the ensemble in the large
is completely determined by the coevolution of the spectral density and a
certain eigenvector correlation function. This coevolution is best visible in
an electrostatic potential of a quaternionic argument built of two complex
variables, the first of which governs standard spectral properties while the
second unravels the hidden dynamics of eigenvector correlation function. We
obtain general large formulas for both spectral density and 1-point
eigenvector correlation function valid for any initial conditions. We exemplify
our studies by solving three examples, and we verify the analytic form of our
solutions with numerical simulations.Comment: 24 pages, 11 figure
Dynamical Isometry is Achieved in Residual Networks in a Universal Way for any Activation Function
We demonstrate that in residual neural networks (ResNets) dynamical isometry
is achievable irrespectively of the activation function used. We do that by
deriving, with the help of Free Probability and Random Matrix Theories, a
universal formula for the spectral density of the input-output Jacobian at
initialization, in the large network width and depth limit. The resulting
singular value spectrum depends on a single parameter, which we calculate for a
variety of popular activation functions, by analyzing the signal propagation in
the artificial neural network. We corroborate our results with numerical
simulations of both random matrices and ResNets applied to the CIFAR-10
classification problem. Moreover, we study the consequence of this universal
behavior for the initial and late phases of the learning processes. We conclude
by drawing attention to the simple fact, that initialization acts as a
confounding factor between the choice of activation function and the rate of
learning. We propose that in ResNets this can be resolved based on our results,
by ensuring the same level of dynamical isometry at initialization
Dysonian dynamics of the Ginibre ensemble
We study the time evolution of Ginibre matrices whose elements undergo
Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the
dynamics of eigenvalues to the evolution of eigenvectors in a non-trivial way,
leading to a system of coupled nonlinear equations resembling those for
turbulent systems. We formulate a mathematical framework allowing simultaneous
description of the flow of eigenvalues and eigenvectors, and we unravel a
hidden dynamics as a function of new complex variable, which in the standard
description is treated as a regulator only. We solve the evolution equations
for large matrices and demonstrate that the non-analytic behavior of the
Green's functions is associated with a shock wave stemming from a Burgers-like
equation describing correlations of eigenvectors. We conjecture that the hidden
dynamics, that we observe for the Ginibre ensemble, is a general feature of
non-Hermitian random matrix models and is relevant to related physical
applications.Comment: 5 pages, 2 figure
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