Combinatorial aspects of Parraud’s asymptotic expansion for GUE matrices

We give a new combinatorial proof of Parraud's formula for the asymptotic expansion in powers of 1/N 2 for the expected trace of polynomials of several independent N N GUE matrices, which expresses the result using a mixture of free difference quotients, introducing new freely independent semicircular variables, and integration with respect to parameters.Our approach streamlines the statement of the formula while clarifying its relationship to the combinatorial genus expansion.] in powers of 1/N 2 (see e.g.Mingo and Speicher (2017, 1.7)).When p is a monomial, the coefficients are given by the enumeration of maps of a certain genus.In physics, the connection between matrix moments and planar maps was studied by 't Hooft (1974a,b), and then generalized to arbitrary genus by Brzin et al. (1978).Harer and Zagier (1986) used this machinery to study the Euler characteristics of moduli spaces of curves.See Zvonkin (1997) for survey of map enumeration and random matrices, and for further applications in random matrices see Ercolani and McLaughlin (2003); Guionnet and Maurel-Segala (2006); Shcherbina (2014).However, due its purely combinatorial nature, the topological expansion is difficult to apply to non-commutative smooth functions beyond the setting of polynomials and power series.Thus, for analytic applications, it is more useful to have an expression for the asymptotic expansion in terms of non-commutative derivatives of the input function, which allows extension of the formula to smooth functions as in the work of Schultz (2005).For instance, to prove sharp results about the convergence