Free Probability Theory and Non-crossing Partitions

Voiculescu's free probability theory { which was introduced in an operator algebraic context, but has since then developed into an exciting theory with a lot of links to other elds { has an interesting combinatorial facet: it can be described by the combinatorial concept of multiplicative functions on the lattice of non-crossing partitions. In this survey I want to explain this connection { without assuming any knowledge neither on free probability theory nor on non-crossing partitions.

[1]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[2]  R. Stanley,et al.  On the foundations of combinatorial theory. VI. The idea of generating function , 1972 .

[3]  Germain Kreweras,et al.  Sur les partitions non croisees d'un cycle , 1972, Discret. Math..

[4]  Paul H. Edelman.  Chain enumeration and non-crossing partitions , 1980, Discret. Math..

[5]  Paul H. EDELMAN.  Multichains, non-crossing partitions and trees , 1982, Discret. Math..

[6]  D. Voiculescu.  Symmetries of some reduced free product C*-algebras , 1985 .

[7]  D. Voiculescu.  Addition of certain non-commuting random variables , 1986 .

[8]  D. Voiculescu.  Limit laws for Random matrices and free products , 1991 .

[9]  R. Speicher.  Free Convolution and the Random Sum of Matrices , 1993 .

[10]  F. Rădulescu.  Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index , 1994 .

[11]  R. Speicher.  Multiplicative functions on the lattice of non-crossing partitions and free convolution , 1994 .

[12]  Dan Voiculescu.  Free Probability Theory: Random Matrices and von Neumann Algebras , 1995 .

[13]  Philippe Biane,et al.  Minimal Factorizations of a Cycle and Central Multiplicative Functions on the Infinite Symmetric Group , 1996, J. Comb. Theory, Ser. A.

[14]  A. Nica.  R-Transforms of Free Joint Distributions and Non-crossing Partitions , 1996 .

[15]  A. Nica,et al.  On the multiplication of free N-tuples of noncommutative random variables , 1996, funct-an/9604011.

[16]  Alexandru Nica,et al.  R-diagonal pairs - a common approach to Haar unitaries and circular elements , 1996 .

[17]  Alexandru Nica,et al.  A “Fourier Transform” for Multiplicative Functions on Non-Crossing Partitions , 1997 .

[18]  Philippe Biane,et al.  Some properties of crossings and partitions , 1997, Discret. Math..

[19]  Roland Speicher,et al.  Combinatorial Theory of the Free Product With Amalgamation and Operator-Valued Free Probability Theory , 1998 .

[20]  Physical Applications of Freeness , .