Planar diagrams for lattice gauge theory on finite tori

TL;DR

This paper proves an all-orders planar perturbation theory equivalence for quenched $U(N)$ lattice gauge models on finite tori, suggesting a convergent expansion before taking the large N limit.

Contribution

It establishes a rigorous $N= fty$ equivalence for quenched $U(N)$ models on finite lattices using double line notation and analyzes convergence properties.

Findings

Proves all-orders equivalence in planar perturbation theory.

Suggests a positive radius of convergence for the expansion.

Uses double line notation to organize lattice momenta and group indices.

Abstract

An $N=\infty$ equivalence among quenched $U(N)$ models on finite lattice tori of $V$ sites is proven to all orders in planar perturbation theory by putting circulant lattice momenta together with group indices on 't Hooft's double lines. Known estimates for the number of order $N^2$ diagrams, $N\gg V$, and the simultaneous presence of UV and IR cutoffs, suggest a positive radius of convergence for the planar perturbative expansion before the limit $N=\infty$ is taken.