Enumeration of maps with the Dumitriu-Edelman model

TL;DR

This paper develops a new $1/N$ and $eta$ expansion for cumulants in the $eta$-ensemble using the Dumitriu-Edelman model, connecting it to labelled maps and providing a bijective relation to previous expansions.

Contribution

It introduces a novel $1/N$ and $eta$ expansion for cumulants based on labelled maps, involving only orientable maps, and relates it to prior work through a new bijective mapping.

Findings

Expansion expressed in terms of labelled maps.

Relation established between new expansion and LaCroix's expansion.

Involves only orientable maps, differing from previous non-orientable map expansions.

Abstract

We give an expansion in $1/N$ and $\beta$ of the cumulants of power sums of the particles of the $\beta$-ensemble. This new expansion is obtained using the tridiagonal model of Dumitriu and Edelman. The coefficients of the expansion are expressed in terms of suitably labelled maps introduced by Bouttier, Fusy, and Guitter. Our expansion is of a different nature than the one obtained by LaCroix in is study of the $b$-conjecture of Goulden and Jackson, and involves only orientable maps. We are able to relate bijectively the first two orders of our expansion to the one of LaCroix using a novel many-to-one mapping that relates suitably labelled planar maps with two minima and maps on the projective plane.