Abstract

We consider the real eigenvalues of the elliptic Ginibre matrix indexed by the non-Hermiticity parameter τ∈[0,1], and present a Harer-Zagier type recursion formula for the even moments in the form of an 11-term recurrence relation. For the symmetric GOE case (τ=1), it reduces to a known 5-term recurrence relation. On the other hand, for the asymmetric cases when τ<1, the recursion formula is new, even in the special case of the well-studied Ginibre ensemble (τ=0), where it reduces to a 3-term recurrence. For the proof, we derive a seventh-order linear differential equation for the moment generating function.

DOI: http://dx.doi.org/10.1016/j.bulsci.2024.103526