Exact solution for mean energy of 2d Dyson gas at β=1

Using the replica method, we calculate the mean Coulomb energy of two-dimensional (2d) Dyson gas (in quadratic potential and for particular value of Coulomb charge β=1). We find a recursive relation on mean energy and obtain its finite N solution in closed form in terms of the hypergeometric function F23. Using this exact solution, we derive the large-N asymptotics (thermodynamical limit) of mean energy.