Discontinuous Percolation Transitions in Epidemic Processes, Surface Depinning in Random Media and Hamiltonian Random Graphs

TL;DR

This paper explores discontinuous percolation transitions in epidemic models, surface depinning, and Hamiltonian random graphs, revealing complex phase behaviors and tricritical points through numerical and theoretical analysis.

Contribution

It demonstrates the occurrence of discontinuous transitions and tricritical points across diverse models, linking epidemic processes, surface depinning, and random graph theories.

Findings

Discontinuous percolation transitions occur in epidemic and random graph models.

Phase diagrams show tricritical points matching field theory predictions.

Hysteresis loops indicate mixed order phase transitions.

Abstract

Discontinuous percolation transitions and the associated tricritical points are manifest in a wide range of both equilibrium and non-equilibrium cooperative phenomena. To demonstrate this, we present and relate the continuous and first order behaviors in two different classes of models: The first are generalized epidemic processes (GEP) that describe in their spatially embedded version - either on or off a regular lattice - compact or fractal cluster growth in random media at zero temperature. A random graph version of GEP is mapped onto a model previously proposed for complex social contagion. We compute detailed phase diagrams and compare our numerical results at the tricritical point in d = 3 with field theory predictions of Janssen et al. [Phys. Rev. E 70, 026114 (2004)]. The second class consists of exponential ("Hamiltonian", or formally equilibrium) random graph models and includes the Strauss and the 2-star model, where 'chemical potentials' control the densities of links, triangles or 2-stars. When the chemical potentials in either graph model are O(logN), the percolation transition can coincide with a first order phase transition in the density of links, making the former also discontinuous. Hysteresis loops can then be of mixed order, with second order behavior for decreasing link fugacity, and a jump (first order) when it increases.