Combinatorics and loop equations for antisymmetrised and Hermitised matrix product ensembles
Abstract
The order m antisymmetrised matrix product ensemble is represented by the product X1T⋯ XmTJXm⋯ X1, where X1,…,Xm are independent real Ginibre matrices and J is the elementary antisymmetric matrix, while the order m Hermitised matrix product ensemble is represented by X1†⋯ Xm† HX1⋯ Xm, where X1,…,Xm are now independent complex Ginibre matrices and H is a Hermitian matrix drawn from the Gaussian unitary ensemble. These ensembles have recently been shown to be related to certain Muttalib--Borodin ensembles and integrals of Harish-Chandra--Itzykson--Zuber type, thereby motivating further investigation into their eigenvalue statistics. In this work, we construct ribbon graphs and constellations that are enumerated by the mixed cumulants of these ensembles and give loop equation characterisations for the generating functions of said cumulants when m=1.
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BibTeX
@misc{dartois2026combinatorics,
title = {{Combinatorics and loop equations for antisymmetrised and Hermitised matrix product ensembles}},
author = {Stephane Dartois and Anas. A Rahman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36332},
eprint = {2609.36332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36332},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Combinatorics and loop equations for antisymmetrised and Hermitised matrix product ensembles
AU - Stephane Dartois
AU - Anas. A Rahman
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36332
UR - https://arxiv.org/abs/2609.36332
AB - The order m antisymmetrised matrix product ensemble is represented by the product X_1^T⋯ X_m^TJX_m⋯ X_1, where X_1,…,X_m are independent real Ginibre matrices and J is the elementary antisymmetric matrix, while the order m Hermitised matrix product ensemble is represented by X_1^†⋯ X_m^† HX_1⋯ X_m, where X_1,…,X_m are now independent complex Ginibre matrices and H is a Hermitian matrix drawn from the Gaussian unitary ensemble. These ensembles have recently been shown to be related to certain Muttalib--Borodin ensembles and integrals of Harish-Chandra--Itzykson--Zuber type, thereby motivating further investigation into their eigenvalue statistics. In this work, we construct ribbon graphs and constellations that are enumerated by the mixed cumulants of these ensembles and give loop equation characterisations for the generating functions of said cumulants when m=1.
ER -