114 research outputs found
Spectral measure of heavy tailed band and covariance random matrices
We study the asymptotic behavior of the appropriately scaled and possibly
perturbed spectral measure of large random real symmetric matrices with
heavy tailed entries. Specifically, consider the N by N symmetric matrix
whose (i,j) entry is where is an infinite array of i.i.d real variables with common
distribution in the domain of attraction of an -stable law,
, and is a deterministic function. For a random diagonal
independent of and with appropriate rescaling , we
prove that the distribution of converges in
mean towards a limiting probability measure which we characterize. As a special
case, we derive and analyze the almost sure limiting spectral density for
empirical covariance matrices with heavy tailed entries.Comment: 31 pages, minor modifications, mainly in the regularity argument for
Theorem 1.3. To appear in Communications in Mathematical Physic
From interacting particle systems to random matrices
In this contribution we consider stochastic growth models in the
Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large
time distribution and processes and their dependence on the class on initial
condition. This means that the scaling exponents do not uniquely determine the
large time surface statistics, but one has to further divide into subclasses.
Some of the fluctuation laws were first discovered in random matrix models.
Moreover, the limit process for curved limit shape turned out to show up in a
dynamical version of hermitian random matrices, but this analogy does not
extend to the case of symmetric matrices. Therefore the connections between
growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor
corrections in scaling of section 2.
Electrical, morphology and structural properties of biodegradable nanocomposite polyvinyl-acetate/ cellulose nanocrystals
In this work, the dielectric properties and the electrical conductivity of polyvinyl acetate (PVAc) polymer doped with cellulose nanocrystals (CNC), extracted from the date palm rachis, are reported. We investigate the filler effect on the molecular mobility of the PVAc polymer chains and the charge transport properties of this material. PVAc/CNC films structure was characterized by powder X-Ray diffraction (XRD), showing the crystalline behavior of the cellulose filler. The dielectric properties were investigated using impedance spectroscopy, in the frequency range of 102–106 Hz and temperatures from 200 to 350 K. A β relaxation, assigned to the motions of the -OCOCH3 side groups, and α relaxation, associated with the glass transition of the PVAc matrix, can be detected.publishe
Metastability and small eigenvalues in Markov chains
In this letter we announce rigorous results that elucidate the relation
between metastable states and low-lying eigenvalues in Markov chains in a much
more general setting and with considerable greater precision as was so far
available. This includes a sharp uncertainty principle relating all low-lying
eigenvalues to mean times of metastable transitions, a relation between the
support of eigenfunctions and the attractor of a metastable state, and sharp
estimates on the convergence of probability distribution of the metastable
transition times to the exponential distribution.Comment: 5pp, AMSTe
FashionBrain Project: A Vision for Understanding Europe's Fashion Data Universe
A core business in the fashion industry is the understanding and
prediction of customer needs and trends. Search engines and social
networks are at the same time a fundamental bridge and a costly
middleman between the customer’s purchase intention and the
retailer. To better exploit Europe’s distinctive characteristics e.g.,
multiple languages, fashion and cultural differences, it is pivotal to
reduce retailers’ dependence to search engines. This goal can be
achieved by harnessing various data channels (manufacturers and
distribution networks, online shops, large retailers, social media,
market observers, call centers, press/magazines etc.) that retailers
can leverage in order to gain more insight about potential buyers,
and on the industry trends as a whole. This can enable the creation
of novel on-line shopping experiences, the detection of influencers,
and the prediction of upcoming fashion trends.
In this paper, we provide an overview of the main research
challenges and an analysis of the most promising technological
solutions that we are investigating in the FashionBrain project
Dynamical ultrametricity in the critical trap model
We show that the trap model at its critical temperature presents dynamical
ultrametricity in the sense of Cugliandolo and Kurchan [CuKu94]. We use the
explicit analytic solution of this model to discuss several issues that arise
in the context of mean-field glassy dynamics, such as the scaling form of the
correlation function, and the finite time (or finite forcing) corrections to
ultrametricity, that are found to decay only logarithmically with the
associated time scale, as well as the fluctuation dissipation ratio. We also
argue that in the multilevel trap model, the short time dynamics is dominated
by the level which is at its critical temperature, so that dynamical
ultrametricity should hold in the whole glassy temperature range. We revisit
some experimental data on spin-glasses in light of these results.Comment: 7 pages, 4 .eps figures. submitted to J. Phys.
Fluctuations for the Ginzburg-Landau Interface Model on a Bounded Domain
We study the massless field on , where is a bounded domain with smooth boundary, with Hamiltonian
\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y)). The interaction \CV is assumed
to be symmetric and uniformly convex. This is a general model for a
-dimensional effective interface where represents the height. We
take our boundary conditions to be a continuous perturbation of a macroscopic
tilt: for , , and
continuous. We prove that the fluctuations of linear
functionals of about the tilt converge in the limit to a Gaussian free
field on , the standard Gaussian with respect to the weighted Dirichlet
inner product for some explicit . In a subsequent article,
we will employ the tools developed here to resolve a conjecture of Sheffield
that the zero contour lines of are asymptotically described by , a
conformally invariant random curve.Comment: 58 page
On small-noise equations with degenerate limiting system arising from volatility models
The one-dimensional SDE with non Lipschitz diffusion coefficient is widely
studied in mathematical finance. Several works have proposed asymptotic
analysis of densities and implied volatilities in models involving instances of
this equation, based on a careful implementation of saddle-point methods and
(essentially) the explicit knowledge of Fourier transforms. Recent research on
tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and
S.~Violante. Marginal density expansions for diffusions and stochastic
volatility, part II: Applications. 2013, arxiv:1305.6765] suggests to work with
the rescaled variable : while
allowing to turn a space asymptotic problem into a small- problem
with fixed terminal point, the process satisfies a SDE in
Wentzell--Freidlin form (i.e. with driving noise ). We prove a
pathwise large deviation principle for the process as
. As it will become clear, the limiting ODE governing the
large deviations admits infinitely many solutions, a non-standard situation in
the Wentzell--Freidlin theory. As for applications, the -scaling
allows to derive exact log-asymptotics for path functionals of the process:
while on the one hand the resulting formulae are confirmed by the CIR-CEV
benchmarks, on the other hand the large deviation approach (i) applies to
equations with a more general drift term and (ii) potentially opens the way to
heat kernel analysis for higher-dimensional diffusions involving such an SDE as
a component.Comment: 21 pages, 1 figur
Time separation as a hidden variable to the Copenhagen school of quantum mechanics
The Bohr radius is a space-like separation between the proton and electron in
the hydrogen atom. According to the Copenhagen school of quantum mechanics, the
proton is sitting in the absolute Lorentz frame. If this hydrogen atom is
observed from a different Lorentz frame, there is a time-like separation
linearly mixed with the Bohr radius. Indeed, the time-separation is one of the
essential variables in high-energy hadronic physics where the hadron is a bound
state of the quarks, while thoroughly hidden in the present form of quantum
mechanics. It will be concluded that this variable is hidden in Feynman's rest
of the universe. It is noted first that Feynman's Lorentz-invariant
differential equation for the bound-state quarks has a set of solutions which
describe all essential features of hadronic physics. These solutions explicitly
depend on the time separation between the quarks. This set also forms the
mathematical basis for two-mode squeezed states in quantum optics, where both
photons are observable, but one of them can be treated a variable hidden in the
rest of the universe. The physics of this two-mode state can then be translated
into the time-separation variable in the quark model. As in the case of the
un-observed photon, the hidden time-separation variable manifests itself as an
increase in entropy and uncertainty.Comment: LaTex 10 pages with 5 figure. Invited paper presented at the
Conference on Advances in Quantum Theory (Vaxjo, Sweden, June 2010), to be
published in one of the AIP Conference Proceedings serie
On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev
inequalities for a class of Boltzmann-Gibbs measures with singular interaction.
Such measures allow to model one-dimensional particles with confinement and
singular pair interaction. The functional inequalities come from convexity. We
prove and characterize optimality in the case of quadratic confinement via a
factorization of the measure. This optimality phenomenon holds for all beta
Hermite ensembles including the Gaussian unitary ensemble, a famous exactly
solvable model of random matrix theory. We further explore exact solvability by
reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting
the Hermite-Lassalle orthogonal polynomials as a complete set of
eigenfunctions. We also discuss the consequence of the log-Sobolev inequality
in terms of concentration of measure for Lipschitz functions such as maxima and
linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional
Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics
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