Critical Properties of the One-Dimensional Forest-Fire Model

TL;DR

This paper investigates the critical properties of a one-dimensional forest-fire model with lightning, combining numerical simulations and analytical methods to uncover new correlation lengths and spectral behaviors near the critical point.

Contribution

It introduces a Hamiltonian formulation for the model, enabling analytical study of the stationary state and relaxation spectrum close to criticality.

Findings

Discovery of a new correlation length with exponent ~5/6.

Numerical analysis of the relaxation spectrum and critical gap.

Simplified models can reproduce critical correlations but not off-critical behavior.

Abstract

The one-dimensional forest-fire model including lightnings is studied numerically and analytically. For the tree correlation function, a new correlation length with critical exponent \nu ~ 5/6 is found by simulations. A Hamiltonian formulation is introduced which enables one to study the stationary state close to the critical point using quantum-mechanical perturbation theory. With this formulation also the structure of the low-lying relaxation spectrum and the critical behaviour of the smallest complex gap are investigated numerically. Finally, it is shown that critical correlation functions can be obtained from a simplified model involving only the total number of trees although such simplified models are unable to reproduce the correct off-critical behaviour.