We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer-Major theorem.Comment: Published in at http://dx.doi.org/10.1214/11-AOP657 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We study the (two-parameter) Segal-Bargmann transform B N s,t on the unitary group U N , for large N . Acting on matrix valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit G s,t as N → ∞, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of trace polynomials, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal-Bargmann transform. We prove several concentration of measure and limit theorems, giving a direct connection from the finite-dimensional transform B N s,t to its limit G s,t . We characterize the operator G s,t through its inverse action on the standard polynomial basis. Finally, we show that, in the case s = t, the limit transform G t,t is the "free Hall transform" G t introduced by Biane.This highlights the fact that the Segal-Bargmann transform does not preserve the space of polynomial functions of a U N -variable; in general, it maps such functions to trace polynomials. Definition 1.7. Let C[u, u −1 ] denote the algebra of Laurent polynomials in a single variable u:C[u, u −1 ] = k∈Z a k u k : a k ∈ C, a k = 0 for all but finitely-many k ,(1.11)

The free multiplicative Brownian motion bt is the large-N limit of the Brownian motion on GL(N ; C), in the sense of * -distributions. The natural candidate for the large-N limit of the empirical distribution of eigenvalues is thus the Brown measure of bt. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region Σt that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density Wt on Σt, which is strictly positive and real analytic on Σt. This density has a simple form in polar coordinates:, where wt is an analytic function determined by the geometry of the region Σt.We show also that the spectral measure of free unitary Brownian motion ut is a "shadow" of the Brown measure of bt, precisely mirroring the relationship between Wigner's semicircle law and Ginibre's circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results. t 4. Properties of Σ t 5. The PDE for S 6. The Hamilton-Jacobi method 6.1. Setting up the method 6.2. Constants of motion 6.3. Solving the equations 6.4. More about the lifetime of the solution 6.5. Surjectivity

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Abstract. We prove an analog of Janson's strong hypercontractivity inequality in a class of non-commutative "holomorphic" algebras. Our setting is the q-Gaussian algebras Γq associated to the q-Fock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a q-Segal-Bargmann transform, and prove Janson's strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer.