127 research outputs found
Large deviations of the maximum of independent and identically distributed random variables
A pedagogical account of some aspects of Extreme Value Statistics (EVS) is
presented from the somewhat non-standard viewpoint of Large Deviation Theory.
We address the following problem: given a set of i.i.d. random variables
drawn from a parent probability density function (pdf)
, what is the probability that the maximum value of the set
is "atypically larger" than expected? The cases
of exponential and Gaussian distributed variables are worked out in detail, and
the right rate function for a general pdf in the Gumbel basin of attraction is
derived. The Gaussian case convincingly demonstrates that the full rate
function cannot be determined from the knowledge of the limiting distribution
(Gumbel) alone, thus implying that it indeed carries additional information.
Given the simplicity and richness of the result and its derivation, its absence
from textbooks, tutorials and lecture notes on EVS for physicists appears
inexplicable.Comment: 14 pag., 1 fig. - Accepted for publication in European Journal of
Physic
Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities
We collect explicit and user-friendly expressions for one-point densities of
the real eigenvalues of Wishart-Laguerre and Jacobi
random matrices with orthogonal, unitary and symplectic symmetry. Using these
formulae, we compute integer moments for all
symmetry classes without any large approximation. In particular, our
results provide exact expressions for moments of transmission eigenvalues in
chaotic cavities with time-reversal or spin-flip symmetry and supporting a
finite and arbitrary number of electronic channels in the two incoming leads.Comment: 27 pages, 3 figures. Typos fixed, references adde
Large deviations of spread measures for Gaussian matrices
For a large Gaussian matrix, we compute the joint statistics,
including large deviation tails, of generalized and total variance - the scaled
log-determinant and trace of the corresponding covariance
matrix. Using a Coulomb gas technique, we find that the Laplace transform of
their joint distribution decays for large (with
fixed) as , where is the Dyson index of the ensemble and
is a -independent large deviation function, which we compute exactly for
any . The corresponding large deviation functions in real space are worked
out and checked with extensive numerical simulations. The results are
complemented with a finite treatment based on the Laguerre-Selberg
integral. The statistics of atypically small log-determinants is shown to be
driven by the split-off of the smallest eigenvalue, leading to an abrupt change
in the large deviation speed.Comment: 20 pages, 3 figures. v4: final versio
Recommended from our members
From Wishart to Jacobi ensembles: Statistical properties and applications
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Sixty years after the works of Wigner and Dyson, Random Matrix
Theory still remains a very active and challenging area of research,
with countless applications in mathematical physics, statistical mechanics and beyond. In this thesis, we focus on rotationally invariant
models where the requirement of independence of matrix elements
is dropped. Some classical examples are the Jacobi and Wishart-Laguerre (or chiral) ensembles, which constitute the core of the present
work. The Wishart-Laguerre ensemble contains covariance matrices
of random data, and represents a very important tool in multivariate
data analysis, with recent applications to finance and telecommunications. We will first consider large deviations of the maximum eigenvalue, providing new analytical results for its large N behavior, and
then a power-law deformation of the classical Wishart-Laguerre ensemble, with possible applications to covariance matrices of financial
data. For the Jacobi matrices, which arise naturally in the quantum
conductance problem, we provide analytical formulas for quantities of
interest for the experiments
Universal transient behavior in large dynamical systems on networks
We analyze how the transient dynamics of large dynamical systems in the
vicinity of a stationary point, modeled by a set of randomly coupled linear
differential equations, depends on the network topology. We characterize the
transient response of a system through the evolution in time of the squared
norm of the state vector, which is averaged over different realizations of the
initial perturbation. We develop a mathematical formalism that computes this
quantity for graphs that are locally tree-like. We show that for unidirectional
networks the theory simplifies and general analytical results can be derived.
For example, we derive analytical expressions for the average squared norm for
random directed graphs with a prescribed degree distribution. These analytical
results reveal that unidirectional systems exhibit a high degree of
universality in the sense that the average squared norm only depends on a
single parameter encoding the average interaction strength between the
individual constituents. In addition, we derive analytical expressions for the
average squared norm for unidirectional systems with fixed diagonal disorder
and with bimodal diagonal disorder. We illustrate these results with numerical
experiments on large random graphs and on real-world networks.Comment: 19 pages, 7 figures. Substantially enlarged version. Submitted to
Physical Review Researc
Invariant sums of random matrices and the onset of level repulsion
We compute analytically the joint probability density of eigenvalues and the
level spacing statistics for an ensemble of random matrices with interesting
features. It is invariant under the standard symmetry groups (orthogonal and
unitary) and yet the interaction between eigenvalues is not Vandermondian. The
ensemble contains real symmetric or complex hermitian matrices of
the form or respectively. The
diagonal matrices
are
constructed from real eigenvalues drawn \emph{independently} from distributions
, while the matrices and are all
orthogonal or unitary. The average is simultaneously
performed over the symmetry group and the joint distribution of
. We focus on the limits i.) and ii.)
, with . In the limit i.), the resulting sum
develops level repulsion even though the original matrices do not feature it,
and classical RMT universality is restored asymptotically. In the limit ii.)
the spacing distribution attains scaling forms that are computed exactly: for
the orthogonal case, we recover the Wigner's surmise, while for the
unitary case an entirely new universal distribution is obtained. Our results
allow to probe analytically the microscopic statistics of the sum of random
matrices that become asymptotically free. We also give an interpretation of
this model in terms of radial random walks in a matrix space. The analytical
results are corroborated by numerical simulations.Comment: 19 pag., 6 fig. - published versio
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