Only during the past weekend, the fires that are besieging Spain burned 16,000 hectares. According to an article in this newspaper, in 2022 140,000 hectares will have been burned, almost seven times more than the annual average for this period. The advance of fire in the forest is difficult to predict, but mathematical models allow us to understand some essential aspects. Specifically, the so-called percolation theory uses cluster formation models connected in random networks to describe the advance of forest fires. In recent years, the French mathematician Hugo Duminil-Copin has made very important contributions to this area, which has earned it one of the 2022 Fields Medals. His workhalfway between physics and mathematics, has revolutionized the field, solving many of the existing problems and extending the theory to previously inaccessible limits.
To the model the spread of a fire, it is key to calculate the probability that the fire will remain isolated or, on the contrary, spread over large areas. In two dimensions, with a very simple analysis, the forest is modeled as a grid, made up of points, which are the trees, and edges, which connect the points. Each edge – each union of two trees – also includes a value, of the probability that the fire will pass between those two trees. If an edge spreads the fire, it is called an open edge. This simplified model does not take time into account and assumes that all trees are identical and independent.
From this model, it is possible to obtain the probability that the fire reaches the center of the forest, which is mathematically equivalent to calculating the probability that a path is formed (that is, a succession of open edges) that communicates the center of the grid with the outside, where the fire is supposed to have started. We can think that the forest is infinitely large and thus simplify the matter to calculate the probability that there is an infinite path with open edges that passes through the center of the forest.
Until the arrival of Duminil-Copin, this field of research was mainly limited to working out the details of the simplified model we have described, called Bernoulli percolation. Is mathematical Construction, introduced in 1957, it makes it possible to calculate a specific value of the probability of fire spreading between trees –called critical probability– from which the risk of the fire reaching the center of the forest greatly increases. Indeed, if the probability of fire contagion is very low (close to zero), it is almost certain that all the paths will be small (finite) and, therefore, the fire will not reach the center of the forest. On the other hand, if it is high enough (close to one), there will almost certainly be infinite paths. The value of the probability in which this phase transition occurs, between the existence or non-existence of infinite paths, is the critical probability.
Bernoulli’s percolation model has a clear limitation: whether an edge is open or closed is independent of the state of the rest of the edges, which is very unrealistic: in a fire, whether or not the fire spreads between two trees does not depend only of these specimens. Duminil-Copin wanted to make the theory more sophisticated, in order to understand the case where this probability is affected by other relatively distant edges. Thus, the refinement of the Duminil-Copin model makes it possible to consider that the probability that a tree ends up spreading the fire depends on the state of nearby trees.
Percolation theory is also used to model the infiltration of water in rocky soil, the spread of certain diseases, the spread of a rumour, the study of ferromagnetism, and much more. Duminil-Copin specialized in this problem of mathematical physics in his doctorate, which he did at the University of Geneva under the supervision of Stanislav Smirnoffalso a Fields medalist.
In that period, he was stuck for months with a problem. While he was thinking about it, swimming in the sea -sports is another of his great hobbies-, he came to an idea that, although it did not work to solve his initial problem, it did allow him to answer an important combinatorial conjecture. The result was published in 2012 in Annals of Mathematicsone of the most important journals in mathematics, and is one of Duminil-Copin’s most cited contributions by the mathematical community.
As the mathematician himself acknowledges, he could not have made these advances, which are now recognized with the Fields Medal, without his collaborators, with whom he would like to have been able to share the award. Duminil-Copin’s generosity and team vision extends to the rest of the scientific community. For example, he considers that writing an article in the clearest and most elegant way possible is a sign of respect towards the other researchers who will spend time studying and using that work.
Alvaro Romaniega He is a predoctoral researcher at the Institute of Mathematical Sciences (ICMAT) and a fellow in Natural Sciences and Technology at the Residencia de Estudiantes.
Timon G Longoria Agate is coordinator of the ICMAT Mathematical Culture Unit.
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.
Edition and coordination: Agate A. Timón G Longoria (ICMAT).
You can follow MATTER in Facebook, Twitter and Instagramor sign up here to receive our weekly newsletter.
#probability #forest #fire #spread