Guillaume Cébron

Free convolution operators and free Hall transform

Abstract We define an extension of the polynomial calculus on a W ⁎ -probability space by introducing an algebra C { X i : 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 GL N ( 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.

The generalized master fields

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 loops 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.

Segal–Bargmann transform: the q-deformation

We give identifications of the q-deformed Segal–Bargmann transform and define the Segal–Bargmann transform on mixed q-Gaussian variables. We prove that, when defined on the random matrix model of Śniady for the q-Gaussian variable, the classical Segal–Bargmann transform converges to the q-deformed Segal–Bargmann transform in the large N limit. We also show that the q-deformed Segal–Bargmann transform can be recovered as a limit of a mixture of classical and free Segal–Bargmann transform.