Phase transitions in atypical systems induced by a condensation transition on graphs

Abstract

Random graphs undergo structural phase transitions that are crucial for dynamical processes and cooperative behavior of models defined on graphs. In this work we investigate the impact of a first-order structural transition on the thermodynamics of the Ising model defined on Erdős-Rényi random graphs, as well as on the eigenvalue distribution of the adjacency matrix of the same graphical model. The structural transition in question yields graph samples exhibiting condensation of degrees, characterized by a large number of nodes having degrees in a narrow interval. Although the condensed graph samples have correlated degrees, we demonstrate that the equations determining the thermodynamics of the Ising model and the eigenvalue distribution of the adjacency matrix both display their usual forms, characteristic of locally treelike ensembles. By solving these equations, we show that the condensation transition induces distinct thermodynamic first-order transitions between the paramagnetic and the ferromagnetic phases of the Ising model. The condensation transition also leads to an abrupt change in the global eigenvalue statistics of the adjacency matrix, which renders the second moment of the eigenvalue distribution discontinuous. As a side result, we derive the critical line determining the percolation transition in Erdõs-Rényi graph samples that feature condensation of degrees.