Condensation and prescaling in spatial Polyakov loop correlations far from equilibrium

The far-from-equilibrium dynamics of spatial Polyakov loop correlations, which provide gauge-invariant observables akin to effective particle numbers for gluon plasmas, are investigated within real-time $\mathrm{SU}(N_c)$ lattice gauge theory at weak couplings and large gluon occupations. The momentum zero mode of these correlations reveals the dynamic formation of a condensate, while at nonzero momenta, energy is transported toward the ultraviolet. We demonstrate that the non-zero momentum dynamics is well described by a direct cascade in terms of gauge-invariant Polyakov loop excitations, exhibiting self-similar prescaling indicative of a nonthermal attractor. This behavior can be analytically understood through perturbation theory for the Polyakov loop correlations and the established dynamics of gauge field correlations. We perform simulations for both $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ gauge groups, providing further consistency checks on the $N_c$-dependence of perturbative expectations. No evidence of an inverse cascade toward lower momenta is found for momenta above the electric screening scale.