ABSTRACT
So far we have studied the convergence of a single sequence of random matrices via the convergence of their ESDs. A natural question is how can we give meaning to joint convergence of several sequences of random matrices? Since matrices are in general non-commutative objects, this needs to be done with care. We need to first introduce the notion of a non-commutative probability space which consists of an algebra A $ \mathcal A $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math9_1.tif"/> and a state ϕ $ \phi $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math9_2.tif"/> on it (which is a linear functional with certain properties). Square matrices of a given order in particular form a natural non-commutative probability space with the state being the average expected trace or simply the average trace. The convergence of a collection of variables from a sequence of non-commutative probability spaces is defined in terms of the convergence of the state of all monomials formed out of this collection.
