Second-order finite free probability

Finite free probability is a new field lying at the intersection of random matrix theory and non-commutative probability. It is called “finite” because unlike traditional free probability, which takes the perspective of operators on infinite-dimensional vector spaces, finite free probability focuses on the study of d × d matrices. Both fields study the behaviour of the eigenvalues of random linear transformations under addition. Finite free probability seeks in particular to characterize random matrices in terms of their (random) characteristic polynomials. I studied the covariance between the coefficients of these polynomials, in order to deepen our knowledge of how random characteristic polynomials fluctuate about their expected values. Focusing on a special case related to random unitary matrices, I applied the representation theory of the unitary group to derive a combinatorial summation expression for the covariance.