10,300 research outputs found
The probability of large deviations for the sum functions of spacings
Let 0=U0,n≤U1,n≤⋯≤Un−1,n≤Un,n=1 be an ordered sample from uniform [0,1] distribution, and Din=Ui,n−Ui−1,n, i=1,2,…,n; n=1,2,…, be their spacings, and let f1n,…,fnn be a set of measurable functions. In this paper, the probabilities of the
moderate and Cramer-type large deviation theorems for statistics
Rn(D)=f1n(nD1n)+⋯+fnn(nDnn) are proved. Application of these theorems for determination of the
intermediate efficiencies of the tests based on Rn(D)-type statistic is presented here too
Surface energy and boundary layers for a chain of atoms at low temperature
We analyze the surface energy and boundary layers for a chain of atoms at low
temperature for an interaction potential of Lennard-Jones type. The pressure
(stress) is assumed small but positive and bounded away from zero, while the
temperature goes to zero. Our main results are: (1) As at fixed positive pressure , the Gibbs measures and
for infinite chains and semi-infinite chains satisfy path large
deviations principles. The rate functions are bulk and surface energy
functionals and
. The minimizer of the surface functional
corresponds to zero temperature boundary layers. (2) The surface correction to
the Gibbs free energy converges to the zero temperature surface energy,
characterized with the help of the minimum of
. (3) The bulk Gibbs measure and Gibbs
free energy can be approximated by their Gaussian counterparts. (4) Bounds on
the decay of correlations are provided, some of them uniform in
A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices
Quantum counterparts of certain simple classical systems can exhibit chaotic
behaviour through the statistics of their energy levels and the irregular
spectra of chaotic systems are modelled by eigenvalues of infinite random
matrices. We use known bounds on the distribution function for eigenvalue
spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real
symmetric matrices and show that gamma distributions, which have an important
uniqueness property, can yield an approximation to the GOE distribution. That
has the advantage that then both chaotic and non chaotic cases fit in the
information geometric framework of the manifold of gamma distributions, which
has been the subject of recent work on neighbourhoods of randomness for general
stochastic systems. Additionally, gamma distributions give approximations, to
eigenvalue spacings for the Gaussian unitary ensemble (GUE) of infinite random
hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite
random hermitian matrices with real quaternionic elements, except near the
origin. Gamma distributions do not precisely model the various analytic systems
discussed here, but some features may be useful in studies of qualitative
generic properties in applications to data from real systems which manifestly
seem to exhibit behaviour reminiscent of near-random processes.Comment: 9 pages, 5 figures, 2 tables, 27 references. Updates version 1 with
data and references from feedback receive
Chaotic quantum dots with strongly correlated electrons
Quantum dots pose a problem where one must confront three obstacles:
randomness, interactions and finite size. Yet it is this confluence that allows
one to make some theoretical advances by invoking three theoretical tools:
Random Matrix theory (RMT), the Renormalization Group (RG) and the 1/N
expansion. Here the reader is introduced to these techniques and shown how they
may be combined to answer a set of questions pertaining to quantum dotsComment: latex file 16 pages 8 figures, to appear in Reviews of Modern Physic
Interacting particle systems at the edge of multilevel Dyson Brownian motions
We study the joint asymptotic behavior of spacings between particles at the
edge of multilevel Dyson Brownian motions, when the number of levels tends to
infinity. Despite the global interactions between particles in multilevel Dyson
Brownian motions, we observe a decoupling phenomenon in the limit: the global
interactions become negligible and only the local interactions remain. The
resulting limiting objects are interacting particle systems which can be
described as Brownian versions of certain totally asymmetric exclusion
processes. This is the first appearance of a particle system with local
interactions in the context of general random matrix models.Comment: 32 pages, 1 figur
Interactions in Chaotic Nanoparticles: Fluctuations in Coulomb Blockade Peak Spacings
We use random matrix models to investigate the ground state energy of
electrons confined to a nanoparticle. Our expression for the energy includes
the charging effect, the single-particle energies, and the residual screened
interactions treated in Hartree-Fock. This model is applicable to chaotic
quantum dots or nanoparticles--in these systems the single-particle statistics
follows random matrix theory at energy scales less than the Thouless energy. We
find the distribution of Coulomb blockade peak spacings first for a large dot
in which the residual interactions can be taken constant: the spacing
fluctuations are of order the mean level separation Delta. Corrections to this
limit are studied using the small parameter 1/(kf L): both the residual
interactions and the effect of the changing confinement on the single-particle
levels produce fluctuations of order Delta/sqrt(kf L). The distributions we
find are significantly more like the experimental results than the simple
constant interaction model.Comment: 17 pages, 4 figures, submitted to Phys. Rev.
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