Bifurcations and convergence of the Martinelli-Parisi expansion in the 2d Potts model

We compute the second order in the Martinelli-Parisi expansion for the 2d Potts model for every number of states q. While we obtain improved estimates for the critical temperatures, the results concerning the fixed-point properties are not satisfactory. We relate the difficulties of the expansion to the presence of a bifurcation in the fixed-point solutions and introduce a rearrangement of the expansion in order to take into account nonlinear effects. As a consequence a better evaluation of qsub(c), where a crossover to first-order transition occurs, is obtained. (orig.)