2,893 research outputs found
Infinitesimal non-crossing cumulants and free probability of type B
Free probabilistic considerations of type B first appeared in a paper by
Biane, Goodman and Nica in 2003. Recently, connections between type B and
infinitesimal free probability were put into evidence by Belinschi and
Shlyakhtenko (arXiv:0903.2721). The interplay between "type B" and
"infinitesimal" is also the object of the present paper. We study infinitesimal
freeness for a family of unital subalgebras A_1, ..., A_k in an infinitesimal
noncommutative probability space (A, phi, phi'), and we introduce a concept of
infinitesimal non-crossing cumulant functionals for (A, phi, phi'), obtained by
taking a formal derivative in the formula for usual non-crossing cumulants. We
prove that the infinitesimal freeness of A_1, ... A_k is equivalent to a
vanishing condition for mixed cumulants; this gives the infinitesimal
counterpart for a theorem of Speicher from "usual" free probability. We show
that the lattices of non-crossing partitions of type B appear in the
combinatorial study of (A, phi, phi'), in the formulas for infinitesimal
cumulants and when describing alternating products of infinitesimally free
random variables. As an application of alternating free products, we observe
the infinitesimal analogue for the well-known fact that freeness is preserved
under compression with a free projection. As another application, we observe
the infinitesimal analogue for a well-known procedure used to construct free
families of free Poisson elements. Finally, we discuss situations when the
freeness of A_1, ..., A_k in (A, phi) can be naturally upgraded to
infinitesimal freeness in (A, phi, phi'), for a suitable choice of a "companion
functional" phi'.Comment: 38 pages, 1 figur
On the degree of rapid decay
A finitely generated group \G equipped with a word-length is said to
satisfy property RD if there are such that, for all non-negative
integers , we have whenever a\in\C\G is
supported on elements of length at most .
We show that, for infinite \G, the degree is at least 1/2.Comment: 6 pages, final versio
Free probability aspect of irreducible meander systems, and some related observations about meanders
We consider the concept of irreducible meandric system introduced by Lando
and Zvonkin. We place this concept in the lattice framework of NC(n). As a
consequence, we show that the even generating function for irreducible meandric
systems is the R-transform of XY, where X and Y are classically (commuting)
independent random variables, and each of X,Y has centred semicircular
distribution of variance 1. Following this point of view, we make some
observations about the symmetric linear functional on polynomials which has
R-transform given by the even generating function for meanders.Comment: Two new remarks in Sections 3 and 5, and an additional referenc
Spectral morphisms, K-theory, and stable ranks
We give a brief account of the interplay between spectral morphisms,
K-theory, and stable ranks in the context of Banach algebras.Comment: 12 pages; to appear in the Proceedings of the Workshop on
Noncommutative Geometry (Fields Institute, Toronto 2008
Multi-variable subordination distributions for free additive convolution
Let k be a positive integer and let D_k denote the space of joint
distributions for k-tuples of selfadjoint elements in C*-probability space. The
paper studies the concept of "subordination distribution of \mu \boxplus \nu
with respect to \nu" for \mu, \nu \in D_k, where \boxplus is the operation of
free additive convolution on D_k. The main tools used in this study are
combinatorial properties of R-transforms for joint distributions and a related
operator model, with operators acting on the full Fock space
Multi-variable subordination turns out to have nice relations to a process of
evolution towards \boxplus-infinite divisibility on D_k that was recently found
by Belinschi and Nica (arXiv:0711.3787). Most notably, one gets better insight
into a connection which this process was known to have with free Brownian
motion.Comment: Minor modifications and reference added to 1st version. 34 pages, no
figure
Homotopical stable ranks for Banach algebras
The connected stable rank and the general stable rank are homotopy invariants
for Banach algebras, whereas the Bass stable rank and the topological stable
rank should be thought of as dimensional invariants. This paper studies the two
homotopical stable ranks, viz. their general properties as well as specific
examples and computations. The picture that emerges is that of a strong
affinity between the homotopical stable ranks, and a marked contrast with the
dimensional ones.Comment: 23 pages, final versio
Unimodular graphs and Eisenstein sums
Motivated in part by combinatorial applications to certain sum-product
phenomena, we introduce unimodular graphs over finite fields and, more
generally, over finite valuation rings. We compute the spectrum of the
unimodular graphs, by using Eisenstein sums associated to unramified extensions
of such rings. We derive an estimate for the number of solutions to the
restricted dot product equation over a finite valuation ring.
Furthermore, our spectral analysis leads to the exact value of the
isoperimetric constant for half of the unimodular graphs. We also compute the
spectrum of Platonic graphs over finite valuation rings, and products of such
rings - e.g., . In particular, we deduce an improved lower
bound for the isoperimetric constant of the Platonic graph over
.Comment: V2: minor revisions. To appear in the Journal of Algebraic
Combinatoric
- …