Phase transition with the Berezinskii-Kosterlitz-Thouless singularity in the Ising model on a growing network

We consider the ferromagnetic Ising model on a highly inhomogeneous network created by a growth process. We find that the phase transition in this system is characterized by the Berezinskii-Kosterlitz-Thouless singularity, although critical fluctuations are absent and the mean-field description is exact. Below this infinite order transition, the magnetization behaves as exp(-const/root T-c-T). We show that the critical point separates the phase with the power-law distribution of the linear response to a local field and the phase where this distribution rapidly decreases. We suggest that this phase transition occurs in a wide range of cooperative models with a strong infinite-range inhomogeneity.