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All genus correlation functions for the hermitian 1-matrix model

B. Eynard

TL;DR

This work reformulates the loop equations of the hermitian $1$-matrix model to compute all correlation functions to arbitrary order in the topological expansion via residues on a hyperelliptic spectral curve. It shows that the $k$-point, genus-$h$ correlators $W_k^{(h)}$ admit a cubic-field-theory–like Feynman-tree expansion on the curve, with residues at branch points furnishing the dynamics. The approach yields explicit recursion for $W_k^{(h)}$, enables practical computations (including explicit $3$- and $4$-point cases), and extends to higher genus and the one-cut genus-zero case, where all objects become rational. The framework links random-matrix observables to geometric data on the spectral curve and suggests connections to tau-functions and potential field-theoretic descriptions, with extensions to the two-matrix model and non-Hermitian variants discussed for future work.

Abstract

We rewrite the loop equations of the hermitian matrix model, in a way which allows to compute all the correlation functions, to all orders in the topological $1/N^2$ expansion, as residues on an hyperelliptical curve. Those residues, can be represented diagrammaticaly as Feynmann graphs of a cubic interaction field theory on the curve.

All genus correlation functions for the hermitian 1-matrix model

TL;DR

This work reformulates the loop equations of the hermitian -matrix model to compute all correlation functions to arbitrary order in the topological expansion via residues on a hyperelliptic spectral curve. It shows that the -point, genus- correlators admit a cubic-field-theory–like Feynman-tree expansion on the curve, with residues at branch points furnishing the dynamics. The approach yields explicit recursion for , enables practical computations (including explicit - and -point cases), and extends to higher genus and the one-cut genus-zero case, where all objects become rational. The framework links random-matrix observables to geometric data on the spectral curve and suggests connections to tau-functions and potential field-theoretic descriptions, with extensions to the two-matrix model and non-Hermitian variants discussed for future work.

Abstract

We rewrite the loop equations of the hermitian matrix model, in a way which allows to compute all the correlation functions, to all orders in the topological expansion, as residues on an hyperelliptical curve. Those residues, can be represented diagrammaticaly as Feynmann graphs of a cubic interaction field theory on the curve.

Paper Structure

This paper contains 26 sections, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Loop functions
  4. Filling fractions
  5. The one-loop function and algebraic geometry
  6. The one--loop function
  7. More notations
  8. Hyperelliptical surfaces
  9. Loop equations
  10. Useful notations
  11. Loop equations
  12. Case k=2
  13. k=3
  14. k larger or equal to 3
  15. Cauchy formula
  16. ...and 11 more sections

Figures (2)

  • Figure 1: The 3 skeleton trees contributing to ${\cal T}_{2}^{(2)}$, i.e. with $k+2h-2=4$ edges.
  • Figure 2: All the possible ways of drawing $k-1=1$ external leg and $h=2$ inner edges, so that the graphs are trivalent, and that inner edges never connect different branches. Notice that all but one graph have symmetry factor $4$, and one has $2$.