## Summary

The Gaussian unitary ensemble is a random matrix model (RMM) for the Wigner law. While random matrices in this model are infinitely
divisible, the Wigner law is infinitely divisible not in the classical but in the free sense. We prove that any variance mixture
of Gaussian distributions -- whether infinitely divisible or not in the classical sense -- admits a RMM of non Gaussian infinitely
divisible random matrices. More generally, it is shown that any mixture of the Wigner law admits a RMM. A key role is played
by the fact that the Gaussian distribution is the mixture of Wigner law with the

## Abstract

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible
distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under
this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models
considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely
divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding
classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations
of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix
models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free
selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ∫_{0}
^{∞}
*e*
^{−1}
*d*Ψ_{t}
^{d}, *d* ≥ 1, where Ψ_{t}
^{d} is a *d* × *d* matrix-valued Lévy process satisfying an *I*
_{log} condition.