Matrix Model Calculations beyond the Spherical Limit

TL;DR

The paper develops an improved iterative scheme to compute higher-genus contributions in the hermitian one-matrix model, delivering explicit genus-two results and a doubled-focus formulation that directly yields the double scaling limit up to genus four. It introduces moments to organize couplings, derives a recursive loop-equation-based procedure for $W_g(p)$ and $F_g$, and demonstrates that in the double scaling limit the Hermitian, Complex, and Kontsevich descriptions converge, establishing a Kontsevich equivalence. It further connects these continuum limits to intersection indices on moduli space and to a discretized moduli-space representation via Kontsevich–Penner, showing how boundary data and reductions are encoded in off-dsl sectors. The work thus provides a coherent bridge between matrix-model topological expansions, continuum moduli-space geometry, and integrable-system frameworks, with explicit computational pipelines for higher genus.

Abstract

We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space.