  We show that large-$N$ $\phi ^4$ theory is not trivial if one accepts the
presence of a tachyon with a truly huge mass, and that it allows exact
calculation. We use it to illustrate how to calculate the exact resummed
pressure at finite temperature and verify that it is infrared and ultraviolet
finite even in the zero-mass case. In 3 dimensions a residual effect of the
resummed infrared divergences is that at low temperature or strong coupling the
leading term in the interaction pressure becomes independent of the coupling
and is 4/5 of the free-field pressure. In 4 dimensions the pressure is
well-defined provided that the temperature is below the tachyon mass. We
examine how rapidly this expansion converges and use our analysis to suggest
how one might reorganise perturbation theory to improve the calculation of the
pressure for the QCD plasma.

High Energy Physics - Phenomenology;High Energy Physics - Theory