Statistical density of particles in one dimensional interaction and Jellium Model

We study a one-dimensional gas of $n$ charged particles confined by a potential and interacting through the Riesz potential or a more general potential. In equilibrium, and for symmetric potential the particles arrange themselves symmetrically around the origin within a finite region. Various models will be studied by modifying both the confining potential and the interaction potential. Focusing on the statistical properties of the system, we analyze the position of the rightmost particle, $x_{\text{max}}$, and show that its typical fluctuations are described by a limiting distribution different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also derive the large deviation functions governing the atypical fluctuations of $x_{\text{max}}$ far from its mean.