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A Theorem on the minimal specific energy for a system with \pm 1 chargedparticles interacting through the Yukawa pair potential v is proved which maystated as follows. Let v be represented by scale mixtures of d-dimensionalEuclid's hat (cutoff at short scale distances) with d\geq 2. For any evennumber of particles n, the interacting energy U_{n} divided by n, attains ann-independent minimum at a configuration with zero net charge and particlepositions collapsed altogether to a point. For any odd number of particles n,the ratio U_{n}/(n-1) attains its minimum value, the same of the previouscases, at the configuration with \pm 1 net charge and particle positionscollapsed to a point. This Theorem is then used to resolve an obstructiveremark of an unpublished paper (Remark 7.5 of \cite{Guidi-Marchetti}) which,whether the standard decomposition of the Yukawa potential into scales wereadopted, would impede a direct proof of the convergence of the Mayer series ofthe two-dimensional Yukawa gas for the inverse temperature in the wholeinterval [4\pi ,8\pi ) of collapse. In the present paper, it is provenconvergence up to the second threshold 6\pi and its given explanations on themechanism that allow it to be extend up to 8\pi . The paper distinguishes thematters concerning stability from those related to convergence of the Mayerseries. In respect to the latter the paper dedicates to the Cauchy majorantemethod applied to the density function of Yukawa gas in the interval ofcollapses. It also dedicates to the proof of the main Theorem and estimates ofthe modified Bessel functions of second kind involved in both representationsof two-dimensional Yukawa potential: standard and scale mixture of the Euclid'shat function.

On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapse