36 research outputs found
What drives transient behaviour in complex systems?
We study transient behaviour in the dynamics of complex systems described by
a set of non-linear ODE's. Destabilizing nature of transient trajectories is
discussed and its connection with the eigenvalue-based linearization procedure.
The complexity is realized as a random matrix drawn from a modified May-Wigner
model. Based on the initial response of the system, we identify a novel
stable-transient regime. We calculate exact abundances of typical and extreme
transient trajectories finding both Gaussian and Tracy-Widom distributions
known in extreme value statistics. We identify degrees of freedom driving
transient behaviour as connected to the eigenvectors and encoded in a
non-orthogonality matrix . We accordingly extend the May-Wigner model to
contain a phase with typical transient trajectories present. An exact norm of
the trajectory is obtained in the vanishing limit where it describes a
normal matrix.Comment: 9 pages, 5 figure
On characteristic polynomials for a generalized chiral random matrix ensemble with a source
We evaluate averages involving characteristic polynomials, inverse
characteristic polynomials and ratios of characteristic polynomials for a
random matrix taken from a -deformed Chiral Gaussian Unitary
Ensemble with an external source . Relation to a recently studied
statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see
Y.V. Fyodorov arXiv:1710.04699, is briefly discussed as a motivation to study
asymptotics of these objects in the case of external source proportional to the
identity matrix. In particular, for an associated 'complex bulk/chiral edge'
scaling regime we retrieve the kernel related to Bessel/Macdonald functions.Comment: published versio
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
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
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
Universal spectral shocks in random matrix theory : lessons for QCD
Following Dyson, we treat the eigenvalues of a random matrix as a system of particles undergoing random walks. The dynamics of large matrices is then well described by fluid dynamical equations. In particular, the inviscid Burgers’ equation is ubiquitous and controls the behavior of the spectral density of large matrices. The solutions to this equation exhibit shocks that we interpret as the edges of the spectrum of eigenvalues. Going beyond the large N limit, we show that the average characteristic polynomial (or the average of the inverse characteristic polynomial) obeys equations that are equivalent to a viscid Burgers’ equation, or equivalently a diffusion equation, with 1∕N playing the role of the viscosity and encoding the entire finite N effects. This approach allows us to recover in an elementary way many results concerning the universal behavior of random matrix theories and to look at QCD spectral features from a new perspective
ID-based, proxy, threshold signature scheme
We propose the proxy threshold signature scheme with the application of elegant construction of verifiable delegating key in the ID-based infrastructure, and also with the bilinear pairings. The protocol satisfies the classical security requirements used in the proxy delegation of signing rights. The description of the system architecture and the possible application of the protocol in edge computing designs is enclosed
Investigating structural and functional aspects of the brain’s criticality in stroke
This paper addresses the question of the brain’s critical dynamics after an injury such as a stroke. It is hypothesized that the healthy brain operates near a phase transition (critical point), which provides optimal conditions for information transmission and responses to inputs. If structural damage could cause the critical point to disappear and thus make self-organized criticality unachievable, it would offer the theoretical explanation for the post-stroke impairment of brain function. In our contribution, however, we demonstrate using network models of the brain, that the dynamics remain critical even after a stroke. In cases where the average size of the second-largest cluster of active nodes, which is one of the commonly used indicators of criticality, shows an anomalous behavior, it results from the loss of integrity of the network, quantifiable within graph theory, and not from genuine non-critical dynamics. We propose a new simple model of an artificial stroke that explains this anomaly. The proposed interpretation of the results is confirmed by an analysis of real connectomes acquired from post-stroke patients and a control group. The results presented refer to neurobiological data; however, the conclusions reached apply to a broad class of complex systems that admit a critical state