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

TL;DR

This paper explores how a condensation-induced structural phase transition in Erdős-Rényi graphs affects the thermodynamics of the Ising model and the eigenvalue distribution of the adjacency matrix, revealing first-order transitions and abrupt spectral changes.

Contribution

It demonstrates the impact of a condensation transition on the thermodynamics and spectral properties of models on random graphs, a novel connection not previously detailed.

Findings

Condensation transition induces first-order phase transitions in the Ising model.

Abrupt change in eigenvalue distribution due to condensation transition.

Derived the critical line for percolation in condensed Erdős-Rényi 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 Erd\"os-R\'enyi 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, characterized by a large number of nodes having degrees in a narrow interval. We show that this 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 Erd\"os-R\'enyi graph samples that feature condensation of degrees.