Testing (asymptotic) scaling in Yang-Mills theories in the large-$N_c$ limit

TEK reduction is a well-established technique that allows single-site simulations of Yang-Mills theory in the large-$N_c$ limit by exploiting volume reduction induced by twisted boundary conditions. We performed simulations for $SU(841)$ for several gauge couplings and applied standard Wilson flow techniques combined with a tree-level improvement methodology to set the lattice scale. The wide range of gauge couplings covered by our simulations allows us to explore a region in the coupling space where our data exhibits asymptotic scaling and perturbation theory could be used to analyze the behaviour of the $\beta$-function. In this talk, I will review the methodology used and go through the main results we obtained, including a determination of the $\mathrm{\Lambda }$-parameter of Yang-Mills theory at large-$N_c$ in $\overline{\text{MS}}$-scheme.