Action-angle variables and phase space formulation of Hermitian matrix models
Abstract
We develop a phase space description of large N Hermitian one matrix models directly from the recursions satisfied by the orthogonal polynomials. In the one-cut phase, the recursion lattice naturally gives rise to a semiclassical action-angle pair, which can be mapped canonically to eigenvalue and momentum variables. We show that the resulting momentum profile is determined by the density of zeros of the orthogonal polynomials, and that the same phase space structure is reproduced from the Wigner transform of the Christoffel-Darboux projector and from the planar spectral curve. The Gaussian model provides a simple consistency check of the proposal. We then extend the construction to the symmetric quartic two-cut phase, where a period-two Jacobi recursion produces two Bloch bands and two disconnected phase space components with actions given by the partial 't Hooft couplings. Finally, we outline the finite-gap generalization appropriate to multicut phases.
Related
- Data-Matrix-Based Overall Goodness-of-Fit Testing for Confirmatory Factor Analysis: In-Sample Prediction Error for Observed Variables
- Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative
- Splitting fields of complex hyperbolic groups: matrix extraction, Hermitian descent, and spectral fields
- A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix
- Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality
BibTeX
@misc{chattopadhyay2026actionangle,
title = {{Action-angle variables and phase space formulation of Hermitian matrix models}},
author = {Arghya Chattopadhyay},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28942},
eprint = {2608.28942},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28942},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Action-angle variables and phase space formulation of Hermitian matrix models
AU - Arghya Chattopadhyay
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28942
UR - https://arxiv.org/abs/2608.28942
AB - We develop a phase space description of large N Hermitian one matrix models directly from the recursions satisfied by the orthogonal polynomials. In the one-cut phase, the recursion lattice naturally gives rise to a semiclassical action-angle pair, which can be mapped canonically to eigenvalue and momentum variables. We show that the resulting momentum profile is determined by the density of zeros of the orthogonal polynomials, and that the same phase space structure is reproduced from the Wigner transform of the Christoffel-Darboux projector and from the planar spectral curve. The Gaussian model provides a simple consistency check of the proposal. We then extend the construction to the symmetric quartic two-cut phase, where a period-two Jacobi recursion produces two Bloch bands and two disconnected phase space components with actions given by the partial 't Hooft couplings. Finally, we outline the finite-gap generalization appropriate to multicut phases.
ER -