Critical behavior of the XY model on uncorrelated and correlated random networks

We numerically study the critical behavior of the XY model on the Erdos-Rényi random graph and a growing random network model, representing the uncorrelated and the correlated random networks, respectively. We also checked the dependence of the critical behavior on the choice of order parameters: the ordinary unweighted and the degree-weighted magnetization. On the Erdos-Rényi random network, the critical behavior of the XY model is found to be of the second order with the estimated exponents consistent with the standard mean-field theory for both order parameters. On the growing random network, on the contrary, we found that the critical behavior is not of the standard mean-field type. Rather, it exhibits behavior reminiscent of that in the infinite-order phase transition for both order parameters, such as the lack of discontinuity in specific heat and the non-divergent susceptibility at the critical point, as observed in the percolation and the Potts models on some growing network models.