A New Derivation of the Finite $N$ Master Loop Equation for Lattice Yang-Mills

We give a new derivation of the finite N master loop equation for lattice Yang -Mills theory with structure group SO(N), U(N) or SU(N). The SO(N) case was initially proved by Chatterjee in [6], and SU(N) was analyzed in a follow-up work by Jafarov [23]. Our approach is based on the Langevin dynamic, an SDE on the manifold of configurations, and yields a simple proof via Ito's formula.