Higher genus correlators for the hermitian matrix model with multiple cuts

TL;DR

This work develops an iterative loop-equation framework for the hermitian one-matrix model with an arbitrary number of cuts, enabling explicit genus-one results and uncovering elliptic-integral boundary conditions that define new universality classes. It provides a complete planar solution and a carefully constructed basis to generate higher-genus correlators, with explicit two-cut results that highlight universality in the planar sector. The double-scaling analysis shows that generic scaling reproduces the one-cut continuum, while special mergings or shrinkings of cuts yield distinct continuum limits, suggesting new critical behaviours beyond the traditional one-cut scenario. The methodology and results open avenues for applying multi-cut loop-equation techniques to related models (e.g., O(n), supereigenvalues, complex matrices) and for exploring richer universality structures in matrix models.

Abstract

An iterative scheme is set up for solving the loop equation of the hermitian one-matrix model with a multi-cut structure. Explicit results are presented for genus one for an arbitrary but finite number of cuts. Due to the complicated form of the boundary conditions, the loop correlators now contain elliptic integrals. This demonstrates the existence of new universality classes for the hermitian matrix model. The two-cut solution is investigated in more detail, including the double-scaling limit. It is shown, that in special cases it differs from the known continuum solution with one cut.

Figures (1)

• Figure 1: The contour of integration ${\cal C} = \cup_{i=1}^s {\cal C}_{i}$