will speak about
“Combinatorics and Asymptotics of Matrix Cumulants, and Connections to Second-Order Freeness ”
The matrix cumulants of Capitaine and Casalsis (’06, ’07, ’08) may be interpreted as the contribution of diagrams in the topological expansion of a matrix integral. This interpretation suggests a further refinement of the quantities into the contributions of individual vertices, and interactions between vertices. I will discuss the asymptotics of these quantities and their connection with the fluctuations of large random matrices and second-order freeness. I will be focusing on the real case. I will also consider the quaternionic case, in which this approach has additional computational benefits. In both these cases, the topological expansion involves non-orientable surfaces (in addition to the oriented surfaces appearing in the complex case). I will discuss some of the combinatorial tools useful in the study of such surfaces.