We define an extension of the polynomial calculus on a W * -probability space by introducing an algebra C{Xi : i ∈ I} which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free convolution. It also permits us to characterize the free Segal-Bargmann transform and the free Hall transform introduced by Biane, in a manner which is closer to classical definitions. Finally, we use this extension of polynomial calculus to prove two asymptotic results on random matrices: the convergence for each fixed time, as N tends to ∞, of the * -distribution of the Brownian motion on the linear group GLN (C) to the * -distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U (N ) to the free Hall transform.

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The master field is the large N limit of the Yang-Mills measure on the Euclidean plane. It can be viewed as a non-commutative process indexed by paths on the plane. We construct and study generalized master fields, called free planar Markovian holonomy fields which are versions of the master field where the law of a simple loop can be as more general as it is possible. We prove that those free planar Markovian holonomy fields can be seen as well as the large N limit of some Markovian holonomy fields on the plane with unitary structure group.

This paper investigates homomorphisms à la Bercovici-Pata between additive and multiplicative convolutions. We also consider their matricial versions which are associated with measures on the space of Hermitian matrices and on the unitary group. The previous results combined with a matricial model of Benaych-Georges and Cabanal-Duvillard allows us to define and study the large N limit of a new matricial model on the unitary group for free multiplicative Lévy processes.Date: 2014. 2010 Mathematics Subject Classification. Primary 15B52, 60B15; secondary 46L54, 60E07.