Free probability is a noncommutative analogue of the probability theory introduced by Dan Voiculescu around 1990 and having some precursors in earlier works on random matrices by Freeman Dyson and others.
D. V. Voiculescu, Free Probability Theory: Random Matrices and von Neumann Algebras, Proc. ICM 1994 pdf
D.V. Voiculescu, K.J. Dykema, A. Nica. Free random variables, CRM Monograph Series, vol. 1, Amer. Math. Soc. 1992.
D.V. Voiculescu, Limit laws for random matrices and free products, Inventiones Mathematicae 104:201–220, 1991.
D.V. Voiculescu, Symmetries of some reduced free product $C^\ast$-algebras, in Operator algebras and their connections with topology and ergodic theory (Bucsteni, 1983), Lecture Notes in Math. 1132, 556–588. Springer 1985.
A. Nica, R. Speicher, Lectures on the combinatorics of free probability, Cambridge University Press, 2006
Terrence Tao, notes on free probability, blog entry
wikipedia free probability
Michael Hartglass, Free product $C^\ast$-algebras associated to graphs, free differentials, and laws of loops, arxiv/1509.02553
Roland Friedrich, John McKay, Formal groups, Witt vectors and free probability, arxiv/1204.6522; Homogeneous Lie groups and quantum probability, arxiv/1506.07089