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 beta = 1). We find a recursive relation on mean energy and obtain its finite N solution in closed form in terms of the hypergeometric function F-3(2). Using this exact solution, we derive the large-N asymptotics (thermodynamical limit) of mean energy. (C) 2011 Elsevier B.V. All rights reserved.