Convergence of the Mayer Series via Cauchy Majorant Method with Application to the Yukawa Gas in the Region of Collapse

We construct majorant functions $\Phi (t,z)$ for the Mayer series of pressure satisfying a nonlinear differential equation of first order which can be solved by the method of characteristics. The domain $| z| <(e\tau) ^{-1}$ of convergence of Mayer series is given by the envelop of characteristic intersections. For non negative potentials we derive an explicit solution in terms of the Lambert $W$ --function which is related to the exponential generating function $T$ of rooted trees as $T(x)=-W(-x)$. For stable potentials the solution is majorized by a non negative potential solution. There are many choices in this case and we combine this freedom together with a Lagrange multiplier to examine the Yukawa gas in the region of collapse. We give, in this paper, a sufficient condition to establish a conjecture of Benfatto, Gallavotti and Nicol\'{o}. For any $\beta \in \lbrack 4\pi, 8\pi)$, the Mayer series with the leading terms of the expansion omitted (how many depending on $\beta $) is shown to be convergent provided an improved stability condition holds. Numerical calculations presented indicate this condition is satisfied if few particles are involved.