Well-posedness of the mean field forest fire age evolution equation

TL;DR

This paper proves the well-posedness of a differential equation modeling the age distribution in a mean field forest fire process, capturing self-organized criticality in large random graphs with lightning strikes.

Contribution

It establishes the existence and uniqueness of solutions to the age evolution equation in the mean field forest fire model, advancing understanding of its large-system limit.

Findings

The differential equation accurately describes the asymptotic age distribution.

The model exhibits self-organized criticality without boundary conditions.

Convergence to the deterministic age process is proven.

Abstract

We prove the well-posedness of a differential equation that describes the evolution of the large-system limit of the empirical age measure in the mean field forest fire model of R\'ath and T\'oth (arXiv:0808.2116). This forest fire model is a random graph process on $n$ vertices, whose dynamics combine the Erd\H{o}s-R\'enyi dynamics with a Poisson rain of lightning strikes. All edges in any connected component are deleted as soon as any of its vertices is struck by lightning. Each vertex has an age, which increases at rate $1$ but is reset to $0$ each time it burns. We consider the asymptotic lightning regime in which the model displays self-organized criticality. Crane, R\'ath and Yeo (arXiv:1811.07981) take the initial state to be an inhomogeneous random graph whose edge probabilities depend on the ages of the vertices. They show that as $n \to \infty$ the empirical age distribution converges as a process to the solution of a deterministic autonomous differential equation. It is a nonlinear age-dependent population dynamics model whose age-specific mortality modulus involves the leading eigenfunction of the branching operator of an associated multitype branching process. The differential equation displays self-organized criticality in the sense that the leading eigenvalue of the branching operator is held at $1$ without this being imposed as a boundary condition.