Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0901.1175

Non-crossing linked partitions, the partial order << on NC(n), and the S-transform

Alexandru Nica
View original source ↗Read source document ↗Export metadata ↓

Abstract

The paper establishes a connection between two recent combinatorial developments in free probability: the non-crossing linked partitions introduced by Dykema in 2007 to study the S-transform, and the partial order << on NC(n) introduced by Belinschi and Nica in 2008 in order to study relations between free and Boolean probability. More precisely, one has a canonical bijection between NCL(n) (the set of all non-crossing linked partitions of {1, ..., n}) and the set {(p,q) | p,q in NC(n), p<<q}. As a consequence of this bijection, one gets an alternative description of Dykema's formula expressing the moments of a noncommutative random variable a in terms of the coefficients of the reciprocal S-transform 1/S_a. Moreover, due to the Boolean features of <<, this formula can be simplified to a form which resembles the moment-cumulant formula from c-free probability.

Explore related subjects

math.OAmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexandru Nica. 2009-01-29. Non-crossing linked partitions, the partial order << on NC(n), and the S-transform. https://arxiv.org/abs/0901.1175

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rings of non-commutative functions and their fields of fractions

Semi-free ideal rings, or semifirs, were introduced by Paul M. Cohn to study universal localizations in the non-commutative setting. We provide new examples of semifirs consisting of analytic functions in several non-commuting variables. These examples arise canonically in free analysis by completing the free algebra in the topology of ``uniform convergence on operator-space balls'' in the non-commutative universe of tuples of square matrices of any finite size. We show, in particular, that the ring of (uniformly) entire non-commutative (NC) functions in $d \in \mathbb{N}$ non-commuting variables, $\scr{O}_d$, is a semifir. Every finitely--generated right (or left) ideal in $\scr{O}_d$ is closed, which yields an analytic extension of G. Bergman's nullstellensatz for the free algebra. Any semifir admits a universal skew field of fractions; applying this to $\scr{O}_d$ yields the universal skew field of ``NC meromorphic expressions", $\scr{M} _d$. We show that any $f \in \scr{M} _d$ has a well-defined domain and evaluations in a large class of stably-finite topological algebras, including finite $C^*$-algebras, extending a result of Cohn for NC rational functions. As an application, we extend the almost sure convergence result of Haagerup and Thorbjörnsen for free polynomials evaluated on tuples of random matrices to the setting of NC meromorphic expressions.

math.OA↗
arXiv2026

Quantum channels on duals of von Neumann algebras in the Schrödinger picture

The theory of quantum channels is traditionally studied either on finite-dimensional state spaces or within the Heisenberg picture as completely positive maps on C^*-algebras. In this paper, we consider quantum channels as completely positive maps on the duals of general von Neumann algebras in the Schrodinger picture. We investigate the construction of such channels through Pettis integrals using representations of topological groups.

math.OA↗
arXiv2026

Infinitesimal Freeness of Wigner Matrices

In this paper, within the framework of real infinitesimal free probability introduced by Cébron and the second author, we compute the real infinitesimal free cumulants of independent complex Wigner matrices. Our approach relies on establishing a combinatorial relation between annular non-crossing partitions and families of directed graphs. As a consequence, we demonstrate that independent complex Wigner matrices are asymptotically real infinitesimally free. In particular, we show (under mild conditions) that a complex Wigner matrix is asymptotically infinitesimally free from its transpose.

math.OA↗