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We consider a one-dimensional classical Coulomb gas of $N$ like-charges in aharmonic potential -- also known as the one-dimensional one-component plasma(1dOCP). We compute analytically the probability distribution of the position$x_{\max}$ of the rightmost charge in the limit of large $N$. We show that thetypical fluctuations of $x_{\max}$ around its mean are described by anon-trivial scaling function, with asymmetric tails. This distribution isdifferent from the Tracy-Widom distribution of $x_{\max}$ for the Dyson'slog-gas. We also compute the large deviation functions of $x_{\max}$ explicitlyand show that the system exhibits a third-order phase transition, as in thelog-gas. Our theoretical predictions are verified numerically.

Exact extremal statistics in the classical $1d$ Coulomb gas