TY - JOUR
T1 - On the moments of moments of random matrices and Ehrhart polynomials
AU - Assiotis, Theodoros
AU - Eriksson, Edward
AU - Ni, Wenqi
N1 - Funding Information:
Acknowledgements We are grateful to the School of Mathematics of the University of Edinburgh for financial support through vacation scholarships. We thank P. Zjaikin for taking part in some initial meetings for this project and are grateful to EPSRC for financial support through a vacation internship. We are grateful to E. Bailey for discussions and useful comments on a first draft of this paper. Finally, we are grateful to the referee for a careful reading of the paper and useful comments and suggestions which have improved the presentation. The tikz code for Fig. 2 was produced using SageMath.
Publisher Copyright:
© 2023 The Author(s)
PY - 2023/8/31
Y1 - 2023/8/31
N2 - There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size N goes to infinity. These quantities depend on two parameters k and q and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size N. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey [7]. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.
AB - There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size N goes to infinity. These quantities depend on two parameters k and q and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size N. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey [7]. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.
KW - Ehrhart-Macdonald reciprocity
KW - Gelfand-Tsetlin polytopes
KW - Moments of characteristic polynomials
KW - Random symplectic matrices
KW - Random unitary matrices
U2 - 10.1016/j.aam.2023.102539
DO - 10.1016/j.aam.2023.102539
M3 - Article
SN - 0196-8858
VL - 149
JO - Advances in Applied Mathematics
JF - Advances in Applied Mathematics
M1 - 102539
ER -