First order phase transition and corrections to its parameters in the   O(N) - model
  The temperature phase transition in the $N$-component scalar field theory with spontaneous symmetry breaking is investigated using the method combining the second Legendre transform and with the consideration of gap equations in the extrema of the free energy. After resummation of all super daisy graphs an effective expansion parameter, $(1/2N)^{1/3}$, appears near $T_c$ for large $N$. The perturbation theory in this parameter accounting consistently for the graphs beyond the super daisies is developed. A certain class of such diagrams dominant in 1/N is calculated perturbatively. Corrections to the characteristics of the phase transition due to these contributions are obtained and turn out to be next-to-leading order as compared to the values derived on the super daisy level and do not alter the type of the phase transition which is weakly first-order. In the limit $N$ goes to infinity the phase transition becomes second order. A comparison with other approaches is done. 
M. Bordag, V. Skalozub