2,637 research outputs found
The Feynman graph representation of convolution semigroups and its applications to L\'{e}vy statistics
We consider the Cauchy problem for a pseudo-differential operator which has a
translation-invariant and analytic symbol. For a certain set of initial
conditions, a formal solution is obtained by a perturbative expansion. The
series so obtained can be re-expressed in terms of generalized Feynman graphs
and Feynman rules. The logarithm of the solution can then be represented by a
series containing only the connected Feynman graphs. Under some conditions, it
is shown that the formal solution uniquely determines the real solution by
means of Borel transforms. The formalism is then applied to probabilistic
L\'{e}vy distributions. Here, the Gaussian part of such a distribution is
re-interpreted as a initial condition and a large diffusion expansion for
L\'{e}vy densities is obtained. It is outlined how this expansion can be used
in statistical problems that involve L\'{e}vy distributions.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ106 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Symmetrization for Quantum Networks: a continuous-time approach
In this paper we propose a continuous-time, dissipative Markov dynamics that
asymptotically drives a network of n-dimensional quantum systems to the set of
states that are invariant under the action of the subsystem permutation group.
The Lindblad-type generator of the dynamics is built with two-body subsystem
swap operators, thus satisfying locality constraints, and preserve symmetric
observables. The potential use of the proposed generator in combination with
local control and measurement actions is illustrated with two applications: the
generation of a global pure state and the estimation of the network size.Comment: submitted to MTNS 201
Stochastic aspects of easy quantum groups
We consider several orthogonal quantum groups satisfying the easiness
assumption axiomatized in our previous paper. For each of them we discuss the
computation of the asymptotic law of Tr(u^k) with respect to the Haar measure,
u being the fundamental representation. For the classical groups O_n, S_n we
recover in this way some well-known results of Diaconis and Shahshahani.Comment: 28 page
Infinitely divisible central probability measures on compact Lie groups---regularity, semigroups and transition kernels
We introduce a class of central symmetric infinitely divisible probability
measures on compact Lie groups by lifting the characteristic exponent from the
real line via the Casimir operator. The class includes Gauss, Laplace and
stable-type measures. We find conditions for such a measure to have a smooth
density and give examples. The Hunt semigroup and generator of convolution
semigroups of measures are represented as pseudo-differential operators. For
sufficiently regular convolution semigroups, the transition kernel has a
tractable Fourier expansion and the density at the neutral element may be
expressed as the trace of the Hunt semigroup. We compute the short time
asymptotics of the density at the neutral element for the Cauchy distribution
on the -torus, on SU(2) and on SO(3), where we find markedly different
behaviour than is the case for the usual heat kernel.Comment: Published in at http://dx.doi.org/10.1214/10-AOP604 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary
We construct a four-parameter family of Markov processes on infinite
Gelfand-Tsetlin schemes that preserve the class of central (Gibbs) measures.
Any process in the family induces a Feller Markov process on the
infinite-dimensional boundary of the Gelfand-Tsetlin graph or, equivalently,
the space of extreme characters of the infinite-dimensional unitary group
U(infinity). The process has a unique invariant distribution which arises as
the decomposing measure in a natural problem of harmonic analysis on
U(infinity) posed in arXiv:math/0109193. As was shown in arXiv:math/0109194,
this measure can also be described as a determinantal point process with a
correlation kernel expressed through the Gauss hypergeometric function
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