$\mathrm{U}(N)$ lattice Yang-Mills in the 't Hooft regime

We establish a mass gap, prove the existence of a unique infinite volume limit, and give a new proof of the large $N$ limit for $\mathrm{U}(N)$ lattice Yang-Mills theory in the 't Hooft regime. These results were previously obtained for $\mathrm{SU}(N)$ and $\mathrm{SO}(N)$ lattice Yang-Mills theories as applications of the mixing of the associated Langevin dynamics, which is verified via the Bakry-\'Emery criterion [SZZ23]. For $\mathrm{U}(N)$, however, this approach fails because its Ricci curvature is not uniformly positive, and as a result the Bakry-\'Emery condition cannot be easily verified. To overcome this obstacle, we recast the $\mathrm{U}(N)$ theory as a random-environment $\mathrm{SU}(N)$ model, where the randomness arises from a $\mathrm{U}(1)$ field, and combine cluster-expansion and Langevin-dynamics techniques to analyze the resulting $\mathrm{U}(1)\times\mathrm{SU}(N)$ model.