AI 中文总结
本文从正交多项式递推出发建立大N厄米矩阵模型相空间描述，明确单割相的作用-角变量结构，扩展至双割相并给出多割相的有限间隙推广，经高斯模型检验方案一致性。
AI 中文摘要
我们直接从正交多项式满足的递推关系出发，建立了大N厄米单矩阵模型的相空间描述。在单割相区域，递推格点自然产生半经典的作用-角对，其可正则映射到本征值与动量变量。我们证明，所得动量分布由正交多项式的零点密度决定，且该相空间结构可通过Christoffel-Darboux投影子的Wigner变换以及平面谱曲线重现，高斯模型为该方案提供了简单的一致性检验。随后，我们将该构造扩展到对称四次双割相区域，其中周期为2的雅可比递推产生两个布洛赫带与两个不连通的相空间分量，其作用量由部分't Hooft耦合给出。最后，我们概述了适用于多割相的有限间隙推广。
英文摘要
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.
Comments40 pages with 4 appendices
