"Free" versions of the Strong Szego Limit Theorem

The Strong Szego Limit Theorem is a theorem about the asymptotics of the determinants of large Toeplitz matrices. We will review the theorem and give a standard reformulation as a probabilistic statement about random unitary matrices. We will then formulate a multivariate generalization of the theorem in this latter form, replacing a single unitary with a system of independent random unitaries. However, the standard proofs of the classical theorem do not generalize. We will indicate some of the key ideas in the proof of the generalized theorem, drawing on random matrix theory, free probability, and noncommutative function theory.