Free probability, Planar algebras, Subfactors and Random Matrices

TL;DR

This paper establishes a connection between planar algebras, non-commutative probability spaces, and random matrix ensembles, providing a new framework to understand symmetries in matrix models and subfactor theory.

Contribution

It introduces a natural non-commutative ring associated with planar algebras that forms a probability space, linking algebraic structures to random matrix symmetries and subfactor constructions.

Findings

Constructs a non-commutative probability space from planar algebras.

Describes random matrix ensembles with specific symmetries.

Provides a canonical subfactor construction from planar algebras.

Abstract

To a planar algebra P in the sense of Jones we associate a natural non- commutative ring, which can be viewed as the ring of non-commutative polynomials in several indeterminates, invariant under a symmetry encoded by P. We show that this ring carries a natural structure of a non-commutative probability space. Non-commutative laws on this space turn out to describe random matrix ensembles possessing special sym- metries. As application, we give a canonical construction of a subfactor and its symmetric enveloping algebra associated to a given planar algebra P. This talk is based on joint work with A. Guionnet and V. Jones.