Yang–Mills Mass Gap in 4D: Lattice SU($N$) and Every Compact Simple Group

We prove that four-dimensional lattice Yang–Mills theory with any compact simple gauge group has a unique infinite-volume state, exponential clustering and an explicit mass gap at fixed lattice spacing, past the curvature limit of the ungauged theory, with the ranges $\beta_W\le0.345$ for SU(2) and $\beta_W\le0.76$ for SU(3). We also prove that one estimate of Dobrushin–Shlosman type in a cube of fixed physical size gives a mass gap in physical units for every continuum limit, and that this estimate, which holds at strong coupling and remains open along the continuum tuning, needs for SU($N$) a cube of side at least of order $(\beta_W/\log\beta_W)^{1/4}$.