Moments of the Gaussian β ensembles and the large-N expansion of the densities

Abstract

The loop equation formalism is used to compute the 1/N expansion of the resolvent for the Gaussian β ensemble up to and including the term at O(N −6 ). This allows the moments of the eigenvalue density to be computed up to and including the 12-th power and the smoothed density to be expanded up to and including the term at O(N −6 ). The latter contain non-integrable singularities at the endpoints of the support -we show how to nonetheless make sense of the average of a sufficiently smooth linear statistic. At the special couplings β = 1, 2 and 4 there are characterisations of both the resolvent and the moments which allows for the corresponding expansions to be extended, in some recursive form at least, to arbitrary order. In this regard we give fifth order linear differential equations for the density and resolvent at β = 1 and 4, which complements the known third order linear differential equations for these quantities at β = 2.2010 Mathematics Subject Classification. 15B52; 60K35; 62E13; 33C45. 1 1 2 [δ(λ − 1) + δ(λ + 1)] −

Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

“…This fact is the well-known genus expansion for Gaussian complex matrices. As observed in [74,Theorem 8], (4.2) can be written in terms of a (terminating) hypergeometric series:…”

Loop equation analysis of the circular β ensembles

“…(24) of [35] and eqs. (3.1) to (3.9) of [44] (where corrections to the two earlier results can be found in the third). …”

“…Analogous dualities for the moments in the Gaussian β ensemble were established using Jack polynomial theory in the study of Dimitriu and Edelman [16], and the corresponding results for the generating functions were given in [44].…”

Moments of the ground state density for the d-dimensional Fermi gas in an harmonic trap

“…where the final equality follows by an application of the geometric series formula and use of (3.17); see [12], [14] and [25] for the case d = 1. In [14], the explicit functional form (albeit with some coefficients specified recursively) of {W ∞,1 (l) (z)} was presented, and we read off in particular that…”

“…Although this as a non-integrable singularity as r → √ 2 − , as noted in [25] it can be integrated against power functions using the Euler beta integral. Doing this, we see that the LHS of (3.17) in the case d = 1, l = 1 agrees with the RHS as specified by (1.16) and (1.19).…”