  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.
