553 research outputs found
Wigner chaos and the fourth moment
We prove that a normalized sequence of multiple Wigner integrals (in a fixed
order of free Wigner chaos) converges in law to the standard semicircular
distribution if and only if the corresponding sequence of fourth moments
converges to 2, the fourth moment of the semicircular law. This extends to the
free probabilistic, setting some recent results by Nualart and Peccati on
characterizations of central limit theorems in a fixed order of Gaussian Wiener
chaos. Our proof is combinatorial, analyzing the relevant noncrossing
partitions that control the moments of the integrals. We can also use these
techniques to distinguish the first order of chaos from all others in terms of
distributions; we then use tools from the free Malliavin calculus to give
quantitative bounds on a distance between different orders of chaos. When
applied to highly symmetric kernels, our results yield a new transfer
principle, connecting central limit theorems in free Wigner chaos to those in
Gaussian Wiener chaos. We use this to prove a new free version of an important
classical theorem, the Breuer-Major theorem.Comment: Published in at http://dx.doi.org/10.1214/11-AOP657 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: Horn hypergeometric functions of two variables
HYPERDIRE is a project devoted to the creation of a set of Mathematica-based
programs for the differential reduction of hypergeometric functions. The
current version allows for manipulations involving the full set of Horn-type
hypergeometric functions of two variables, including 30 functions.Comment: minor corrections, publushed versio
compression, traveling salesmen, and stable walks
We show that if is a group of polynomial growth whose growth rate is at
least quadratic then the compression of the wreath product \Z\bwr H
equals . We also show that the compression of
\Z\bwr \Z equals and the compression of
(\Z\bwr\Z)_0 (the zero section of \Z\bwr \Z, equipped with the metric
induced from \Z\bwr \Z) equals . The fact that
the Hilbert compression exponent of \Z\bwr\Z equals while the
Hilbert compression exponent of (\Z\bwr\Z)_0 equals is used to show
that there exists a Lipschitz function f:(\Z\bwr\Z)_0\to L_2 which cannot be
extended to a Lipschitz function defined on all of \Z\bwr \Z
Symmetry and Integrability of Non-Singlet Sectors in Matrix Quantum Mechanics
We study the non-singlet sectors of matrix quantum mechanics (MQM) through an
operator algebra which generates the spectrum. The algebra is a nonlinear
extension of the W_\infty algebra where the nonlinearity comes from the angular
part of the matrix which can not be neglected in the non-singlet sector. The
algebra contains an infinite set of commuting generators which can be regarded
as the conserved currents of MQM. We derive the spectrum and the eigenfunctions
of these conserved quantities by a group theoretical method. An interesting
feature of the spectrum of these charges in the non-singlet sectors is that
they are identical to those of the singlet sector except for the
multiplicities. We also derive the explicit form of these commuting charges in
terms of the eigenvalues of the matrix and show that the interaction terms
which are typical in Calogero-Sutherland system appear. Finally we discuss the
bosonization and rewrite the commuting charges in terms of a free boson
together with a finite number of extra degrees of freedom for the non-singlet
sectors.Comment: 34 pages, 4 figure
Complex projective structures with Schottky holonomy
A Schottky group in PSL(2, C) induces an open hyperbolic handlebody and its
ideal boundary is a closed orientable surface S whose genus is equal to the
rank of the Schottky group. This boundary surface is equipped with a (complex)
projective structure and its holonomy representation is an epimorphism from
pi_1(S) to the Schottky group. We will show that an arbitrary projective
structure with the same holonomy representation is obtained by (2 pi-)grafting
the basic structure described above.Comment: 52 pages, 14 figure
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