  The high temperature expansion is an analytical tool to study critical
phenomena in statistical mechanics. We apply this method to 3d effective
theories of Polyakov loops, which have been derived from 4d lattice Yang-Mills
by means of resummed strong coupling expansions. In particular, the Polyakov
loop susceptibility is computed as a power series in the effective couplings. A
Pad\'e analysis then provides the location of the phase transition in the
effective 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 the
case 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\'e
analysis. The method can be generalized to include fermions and finite density.
