Wildfire data show a clear pattern world wide: the distribution of fire sizes is heavy-tailed. These heavy-tailed size distributions can be fitted with power-laws, enabling the construction of fire risk maps in a manner analogous to that done for earth-quakes. While this provides fire management with useful information, we still lack a detailed mechanistic understanding of how these heavy-tailed distributions arise. Most existing fire models are complex due to their purpose of predicting the spread of a specific fire. This task requires detailed information about local conditions in vegetation, topography and fuel moisture that influence fire spread processes on small scales. However, this complexity and high information demand of predictive models limit our understanding of their statistical properties. More parsimonious models have a severe limitation: although heavy-tailed distributions are obtained, the steepness of this distribution does not vary as in the data. It has been shown that variation can be obtained by assuming different burning conditions for each fire. This assumption, however, is challenged by studies suggesting the existence of nonlinearities and thresholds in the effect of environmental factors, such as temperature, on the area burned by a fire.
We present a model that analytically predicts a power-law distribution of fire sizes of exponent -1.5 for a critical value. We argue that the main process controlling the fire size is the stochastic dynamics of the fire front. The range of steepness (exponent -1.9 to -1.3) observed in the data is obtained by varying the average success of a fire front to travel across the fuel. We conclude that stochasticity during the spread of a fire is able to generate the observed fire size distributions.