502 research outputs found
Fock factorizations, and decompositions of the spaces over general Levy processes
We explicitly construct and study an isometry between the spaces of square
integrable functionals of an arbitrary Levy process and a vector-valued
Gaussian white noise. In particular, we obtain explicit formulas for this
isometry at the level of multiplicative functionals and at the level of
orthogonal decompositions, as well as find its kernel. We consider in detail
the central special case: the isometry between the spaces over a Poisson
process and the corresponding white noise. The key role in our considerations
is played by the notion of measure and Hilbert factorizations and related
notions of multiplicative and additive functionals and logarithm. The obtained
results allow us to introduce a canonical Fock structure (an analogue of the
Wiener--Ito decomposition) in the space over an arbitrary Levy process.
An application to the representation theory of current groups is considered. An
example of a non-Fock factorization is given.Comment: 35 pages; LaTeX; to appear in Russian Math. Survey
Gaussian fluctuations of Young diagrams and structure constants of Jack characters
In this paper, we consider a deformation of Plancherel measure linked to Jack
polynomials. Our main result is the description of the first and second-order
asymptotics of the bulk of a random Young diagram under this distribution,
which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the
first order asymptotics) and Kerov (for the second order asymptotics). This
gives more evidence of the connection with Gaussian -ensemble, already
suggested by some work of Matsumoto.
Our main tool is a polynomiality result for the structure constant of some
quantities that we call Jack characters, recently introduced by Lassalle. We
believe that this result is also interested in itself and we give several other
applications of it.Comment: 71 pages. Minor modifications from version 1. An extended abstract of
this work, with significantly fewer results and a different title, is
available as arXiv:1201.180
A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics
A ``hybrid method'', dedicated to asymptotic coefficient extraction in
combinatorial generating functions, is presented, which combines Darboux's
method and singularity analysis theory. This hybrid method applies to functions
that remain of moderate growth near the unit circle and satisfy suitable
smoothness assumptions--this, even in the case when the unit circle is a
natural boundary. A prime application is to coefficients of several types of
infinite product generating functions, for which full asymptotic expansions
(involving periodic fluctuations at higher orders) can be derived. Examples
relative to permutations, trees, and polynomials over finite fields are treated
in this way.Comment: 31 page
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