Renormalization group approach to the critical behavior of the forest fire model

TL;DR

This paper applies a Renormalization group approach to forest-fire models in one and two dimensions, analytically deriving critical exponents and confirming their accuracy with numerical simulations.

Contribution

It introduces a Renormalization scheme for forest-fire models, identifying a relevant scaling field and analytically computing critical exponents in different dimensions.

Findings

Existence of a relevant scaling field with a repulsive fixed point.

Analytical critical exponents match numerical results.

Model is critical with a tunable control parameter.

Abstract

We introduce a Renormalization scheme for the one and two dimensional Forest-Fire models in order to characterize the nature of the critical state and its scale invariant dynamics. We show the existence of a relevant scaling field associated with a repulsive fixed point. This model is therefore critical in the usual sense because the control parameter has to be tuned to its critical value in order to get criticality. It turns out that this is not just the condition for a time scale separation. The critical exponents are computed analytically and we obtain $\nu=1.0$, $\tau=1.0$ and $\nu=0.65$, $\tau=1.16$ respectively for the one and two dimensional case, in very good agreement with numerical simulations.