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Oblatum 25- VI-1990 In [11][12][13][14][15] we studied a kind of probability theory in the framework of operator algebras, with the tensor product replaced by the free product. The notion corresponding to independent random variables is then, that of free random variables. Examples of pairs of free random variables are provided by convolution operators on a free product group G = G1 * G2 such that the convolutors are supported in G1 and respectively G2. We proved a central limit theorem for free random variables, the limit law being a semi-circle law. We also gave explicit formulae for free convolution which plays the same role for free random variables as usual convolution for independent random variables (see [15] for a survey).The starting point for this paper was to show that the occurrence of the semi-circle law both in the study of free products and of random matrices [17,18,1,5,6,9,16] is not a mere coincidence. We prove here that free random variables naturally arise as limits of random matrices and that Wigner's semicircle law is a consequence of the central limit theorem for free random variables. Actually in this way we obtain a non-commutative limit distribution of a general gaussian random matrix (i.e. no restriction of selfadjointness is imposed) as an operator in a certain operator algebra, Wigner's law being given by the trace of the spectral measure of the selfadjoint component of this operator.The main phenomenon underlying our results is, roughly stated, that letting the size of the matrices go to oo, two matrices the entries of which, taken together, form a family of independent gaussian random variables will have as limit a pair of non-commutative free random variables. We derive also corresponding results for unitary matrices and projections.The paper has 3 sections:Section 1 contains for the readers' convenience preliminary material on free random variables from our papers 1-11-15].Section 2 proves the main result on independent random matrices giving rise asymptotically to free random variables.* Research supported in part by a grant form the National Science Foundation. 202D. Voiculescu -Section 3 deals with consequences of the result in section 2. Among these are the non-commutative limit distribution of a gaussian random matrix, generalizing Wigner's law and the free random variables arising from unitary groups and grassmannians. We also show, how certain asymptotic distributions like those studied in [16] have a natural explanation via free convolution.

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