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The high temperature expansion is an analytical tool to study criticalphenomena in statistical mechanics. We apply this method to 3d effectivetheories of Polyakov loops, which have been derived from 4d lattice Yang-Millsby means of resummed strong coupling expansions. In particular, the Polyakovloop susceptibility is computed as a power series in the effective couplings. APad\'e analysis then provides the location of the phase transition in theeffective theory, which can be mapped back to the parameters of 4d Yang-Mills.Our purely analytical results for the critical couplings $\beta_c(N_\tau)$agree to better than $10\%$ with those from Monte Carlo simulations. For thecase of $SU(2)$, also the critical exponent $\gamma$ is predicted accurately,while a first-order nature as for $SU(3)$ cannot be identified by a Pad\'eanalysis. The method can be generalized to include fermions and finite density.

The Yang-Mills deconfinement transition from a high temperature expansion