  We investigate the Langevin dynamics of various lattice formulations of the
Yang-Mills-Higgs model, where the Higgs component takes values in
$\mathbb{R}^N$, $\mathbb{S}^{N-1}$ or a Lie group. We prove the exponential
ergodicity of the dynamics on the whole lattice via functional inequalities. As
an application, we establish that correlations for a broad range of observables
decay exponentially. Specifically, the infinite volume measure exhibits a
strictly positive mass gap under strong coupling conditions. Moreover,
appropriately rescaled observables exhibit factorized correlations in the large
$N$ limit when the state space is compact. Our approach involves disintegration
and a nuanced analysis of correlations to effectively control the unbounded
Higgs component.
