  Using the path integral method, we calculate the partition function and
generating functional (of the field strengths) on the nonlocal generalized 2D
Yang - Mills theories ($nlgYM_2$'s), which is nonlocal in auxiliary field [14].
Our calculations is done for general surfaces. We find a general expression for
free energy of $W(\phi) = \phi^{2k}$ in $nlgYM_2$ theories at the strong
coupling phase (SCP) regime ($A > A_c$) for large groups. In the specific
$\phi^4$ model, we show that the theory has a third order phase transition.
