Transition from weak to strong coupling in thermal gauge theories: Lessons from $\mathcal{N}=4$ SYM Theory

We investigate the transition between weak and strong coupling in thermal $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory as a function of 't Hooft coupling $\lambda$ for several quantities of phenomenological interest for which next-to-leading order calculations are available in both regimes. We use Padé approximants to interpolate between the weak and strong coupling expansions and determine the location and width of the transition between the asymptotic regimes.