Figure 1

Propagation of fires in the modified DS model. Circles indicate sites occupied by $A$ (white) or $B$ [gray (green online)] trees. (a) On some initial configuration, lightning strikes the site indicated by the arrow. (b) The state after the instantaneous burning of the connected set of $A$ trees. The two $B$ trees that are highlighted as black have become “active.” (c) One of the active $B$ sites is chosen at random and propagates fire to its neighbor trees. Additional $B$ sites may become active. (d) Final stage when there are no more active $B$ sites. Along the whole process, fires of sizes 5, 4, 4, and 6 have occurred (contours are highlighted). The total size of the burnt cluster is thus 19.Reuse & Permissions

Figure 2

(a) Temporal sequence of fires, in the modified DS model on a system of size $4000\times 4000$, with $r=5\times {10}^4$ and $w={10}^{-3}$. Events that occur at the same value of $t/t_0$ form what I have called a cluster, corresponding to the initial fire, generated by lightning, and all secondary fires propagated through $B$ trees. In (b), the same events are plotted as a function of its internal time within the cluster, set to 0 for the first event of the cluster. The continuous line is the average AS rate.Reuse & Permissions

Figure 3

Statistics of events for simulations with $r={10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ (from left to right) and $w=0.01$ (system size is up to $20000\times 20000$). Lower curves correspond to individual events, whereas the upper ones are the results for the complete clusters. The statistics of clusters coincides with the result for single events in the case $w=0$.Reuse & Permissions

Figure 4

(a) Partial size distribution of events in the modified DS model ($r={10}^5$, $w=0.01$), restricted to the indicated events in the clusters. The distribution has an exponent close to that of the original model for the initial event and becomes progressively steeper as successive secondary events are considered. (b) An actual example of this phenomenon (details are provided in Supplemental Material [18]). The lower points display the distribution of EQs in Southern California during a 20-year time period. The upper curves correspond to the distributions of events that occurred in the two time windows indicated, after EQs of magnitude 4.5 or larger. Note that these two partial distribution are limited to magnitudes lower than 4.5, since larger events are counted as main shocks. The behavior is qualitatively similar to that in (a). Straight lines are for reference only.Reuse & Permissions