  Using a family of graded algebra structures on a planar algebra and a family
of traces coming from random matrix theory, we obtain a tower of
non-commutative probability spaces, naturally associated to a given planar
algebra. The associated von Neumann algebras are II$_{1}$ factors whose
inclusions realize the given planar algebra as a system of higher relative
commutants. We thus give an alternative proof to a result of Popa that every
planar algebra can be realized by a subfactor.
