We consider a one-dimensional classical Coulomb gas of N like-charges in a harmonic potential – also known as the one-dimensional one-component plasma (1dOCP). We compute analytically the probability distribution of the position xmax of the rightmost charge in the limit of large N. We show that the typical fluctuations of xmax around its mean are described by a non- trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of xmax for the Dyson’s log-gas. We also compute the large deviation functions of xmax explicitly and show that the system exhibits a third-order phase transition, as in the log-gas. I’ll also discuss some results on the distribution of the index, i.e., the
number of charges on the positive semi-axis.