A remark on the phase transitions of modified action spin and gauge models

We consider the phase diagrams of modified action gauge and spin models and concentrate on their periphery - infinitely far from their origins (zero temperature - β^-1 = 0). In this limit the exact positions of the phase transitions are found by looking for the global minimum of the single plaquette action (for a spin system - the single link energy). As the parameters of the model are varied, the position of such a global minimum is in general changed. When this changed is non-analytic, a phase transition takes place. The phase structure for finite β is clearly similar, but not identical to the infinite β one. We discuss several finite β corrections that should be applied to the exactly known infinite β picture. We confront our analysis for infinite β² = ∑_iβ²_i with the Monte Carlo simulations for two four-dimensional gauge systems: an SU(3) gauge model with action S=-Re∑_p(β₁trU_p+β₂(tr U_p)²) and an SU(2) model with S=-Re Σ_p[β₁ 1 2trU_p+β₂( 1 2trU_p)²+β₃( 1 2trU _p)³].