28,484 research outputs found
The Ginibre ensemble and Gaussian analytic functions
We show that as changes, the characteristic polynomial of the
random matrix with i.i.d. complex Gaussian entries can be described recursively
through a process analogous to P\'olya's urn scheme. As a result, we get a
random analytic function in the limit, which is given by a mixture of Gaussian
analytic functions. This gives another reason why the zeros of Gaussian
analytic functions and the Ginibre ensemble exhibit similar local repulsion,
but different global behavior. Our approach gives new explicit formulas for the
limiting analytic function.Comment: 23 pages, 1 figur
Multiplying a Gaussian Matrix by a Gaussian Vector
We provide a new and simple characterization of the multivariate generalized
Laplace distribution. In particular, this result implies that the product of a
Gaussian matrix with independent and identically distributed columns by an
independent isotropic Gaussian vector follows a symmetric multivariate
generalized Laplace distribution
Real eigenvalues of non-Gaussian random matrices and their products
We study the properties of the eigenvalues of real random matrices and their
products. It is known that when the matrix elements are Gaussian-distributed
independent random variables, the fraction of real eigenvalues tends to unity
as the number of matrices in the product increases. Here we present numerical
evidence that this phenomenon is robust with respect to the probability
distribution of matrix elements, and is therefore a general property that
merits detailed investigation. Since the elements of the product matrix are no
longer distributed as those of the single matrix nor they remain independent
random variables, we study the role of these two factors in detail. We study
numerically the properties of the Hadamard (or Schur) product of matrices and
also the product of matrices whose entries are independent but have the same
marginal distribution as that of normal products of matrices, and find that
under repeated multiplication, the probability of all eigenvalues to be real
increases in both cases, but saturates to a constant below unity showing that
the correlations amongst the matrix elements are responsible for the approach
to one. To investigate the role of the non-normal nature of the probability
distributions, we present a thorough analytical treatment of the
single matrix for several standard distributions. Within the class of smooth
distributions with zero mean and finite variance, our results indicate that the
Gaussian distribution has the maximum probability of real eigenvalues, but the
Cauchy distribution characterised by infinite variance is found to have a
larger probability of real eigenvalues than the normal. We also find that for
the two-dimensional single matrices, the probability of real eigenvalues lies
in the range [5/8,7/8].Comment: To appear in J. Phys. A: Math, Theo
Stochastic expansions using continuous dictionaries: L\'{e}vy adaptive regression kernels
This article describes a new class of prior distributions for nonparametric
function estimation. The unknown function is modeled as a limit of weighted
sums of kernels or generator functions indexed by continuous parameters that
control local and global features such as their translation, dilation,
modulation and shape. L\'{e}vy random fields and their stochastic integrals are
employed to induce prior distributions for the unknown functions or,
equivalently, for the number of kernels and for the parameters governing their
features. Scaling, shape, and other features of the generating functions are
location-specific to allow quite different function properties in different
parts of the space, as with wavelet bases and other methods employing
overcomplete dictionaries. We provide conditions under which the stochastic
expansions converge in specified Besov or Sobolev norms. Under a Gaussian error
model, this may be viewed as a sparse regression problem, with regularization
induced via the L\'{e}vy random field prior distribution. Posterior inference
for the unknown functions is based on a reversible jump Markov chain Monte
Carlo algorithm. We compare the L\'{e}vy Adaptive Regression Kernel (LARK)
method to wavelet-based methods using some of the standard test functions, and
illustrate its flexibility and adaptability in nonstationary applications.Comment: Published in at http://dx.doi.org/10.1214/11-AOS889 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Irreducible decomposition of Gaussian distributions and the spectrum of black-body radiation
It is shown that the energy of a mode of a classical chaotic field, following
the continuous exponential distribution as a classical random variable, can be
uniquely decomposed into a sum of its fractional part and of its integer part.
The integer part is a discrete random variable (we call it Planck variable)
whose distribution is just the Bose distribution yielding the Planck law of
black-body radiation. The fractional part is the dark part (we call is dark
variable) with a continuous distribution, which is, of course, not observed in
the experiments. It is proved that the Bose distribution is infinitely
divisible, and the irreducible decomposition of it is given. The Planck
variable can be decomposed into an infinite sum of independent binary random
variables representing the binary photons (more accurately photo-molecules or
photo-multiplets) of energies 2^s*h*nu with s=0,1,2... . These binary photons
follow the Fermi statistics. Consequently, the black-body radiation can be
viewed as a mixture of statistically and thermodynamically independent fermion
gases consisting of binary photons. The binary photons give a natural tool for
the dyadic expansion of arbitrary (but not coherent) ordinary photon
excitations. It is shown that the binary photons have wave-particle
fluctuations of fermions. These fluctuations combine to give the wave-particle
fluctuations of the original bosonic photons expressed by the Einstein
fluctuation formula.Comment: 29 page
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