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 \cite{Cha}, and $SU(N)$ was analyzed in a follow-up work by Jafarov \cite{Jafar}. Our approach is based on the Langevin dynamic, an SDE on the manifold of configurations, and yields a simple proof via It\^o's formula.