A combinatorial approach to free probability theory has been developped by Roland Speicher, based on the notion of noncrossing cumulants, a free analogue of the classical theory of cumulants in a probability theory. We review this theory, and explain the connections between free probability theory and random matrices. We relate noncrossing cumulants to classical cumulants and also to characters of large symmetric groups. Finally we give applications to the asymptotics of representations of symmetric groups, specifically to the Littlewood-Richardson rule.
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