Convergence Of The Phase Space Cluster Expansion For Lattice Yang-Mills Theory

We provide a proof for the absolute convergence of the phase space cluster expansion for a lattice-regularized Yang-Mills theory with a compact, simple Lie group. The theoryis mapped to a statistical mechanical system of interacting polymers, and we prove convergencefor sufficiently small bare couplings using the Kotecký-Preiss criterion. The core of the proof isa combinatorial bound on the sum over all Feynman diagram topologies, which circumvents thefactorial growth of naive perturbation theory. As a result, and under a standard assumption ofvacuum stability, we establish uniform bounds on the connected Schwinger functions, controlledby a weighted norm on the space of continuum action functionals. This result provides a keyanalytic tool for the non-perturbative construction of the quantum theory.