Last weekend alone, the fires besieging Spain burned 16,000 hectares. According to an article in this newspaper, 140,000 hectares will have been burned by 2022, almost seven times the annual average for that period. The advance of fire in the forest is difficult to predict, but mathematical models allow us to understand some key aspects. In particular, the so-called percolation theory uses models of cumulus formation connected in random networks to describe wildfire progression. In recent years, French mathematician Hugo Duminil-Copin has made extremely important contributions to this field, earning it one of the 2022 Fields Medals. His work, which combines physics and mathematics, has revolutionized the field, solving and expanding many of the existing problems the theory to hitherto inaccessible limits.
When modeling the spread of a fire, it is crucial 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 that are the trees and edges that connect the points. Each edge – each union of two trees – also contains a value for 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 is time-insensitive and assumes that all trees are identical and independent.
modeling of a forest; the trees are represented by the grid points and the open borders by lines connecting the points. You can see red-edged paths appearing where the fire could pass.
From this model it is possible to obtain the probability that the fire will reach the center of the forest, which is mathematically equivalent to calculating the probability that a path (i.e. a sequence of open edges) will form that communicates the center of the grid with the outside where the fire is said to have started. We can think of the forest as infinitely large, and so simplify things to calculate the probability that there is an infinite open-edged path through the middle of the forest.
Until the arrival of Duminil-Copin, this area of research was mainly limited to elaborating the details of the simplified model we have described, called Bernoulli percolation. This mathematical construction, introduced in 1957, makes it possible to calculate a certain value for the probability of a fire spreading between trees – the so-called critical probability – above which the risk of the fire reaching the center of the forest increases sharply. If the probability of fire contagion is very small (near zero) then it is almost certain that all paths are small (finite) and therefore the fire will not reach the middle of the forest. On the other hand, if it’s high enough (close to one), there will almost certainly be infinite paths. The value of the probability that 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 other edges, which is very unrealistic: in the case of a fire, it does not only depend on whether the fire spreads between two trees or not these specimens. Duminil-Copin wanted to refine the theory 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 of a tree spreading fire depends on the state of neighboring trees.
Settings for Bernoulli percolation with probabilities below (left) and above (right) the critical parameter. In the second case, an infinite path appears.
Percolation theory is also used to model water intrusion into rocky ground, the spread of certain diseases, the spread of a rumor, the study of ferromagnetism, and much more. Duminil-Copin specialized in this problem of mathematical physics in his doctoral studies at the University of Geneva under the supervision of Stanislav Smirnov, also a Fields Medalist.
During that time, he was stuck with a problem for months. While contemplating swimming in the sea—sport is another of his great hobbies—an idea occurred to him that, while not solving his original problem, allowed him to solve an important conjecture in combinatorics. The result was published in 2012 in the Annals of Mathematics, one of the most important journals for mathematics, and is one of Duminil-Copin’s most cited contributions in the mathematical community.
As the mathematician himself acknowledges, he could not have made the advances now recognized by the Fields Medal without his collaborators, with whom he would have happily shared the award. Duminil-Copin’s generosity and team vision extends to the rest of the scientific community. For example, consider writing an article as clearly and elegantly as possible as a mark of respect for other researchers who spend time studying and using your work.
Alvaro Romaniega He is a PhD 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 the coordinator of the ICMAT Mathematical Culture Unit.
coffee and theorems is a section dedicated to mathematics and the environment in which it arises, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline and meeting points between mathematics and others share social networks and cultural expressions and remember those who shaped its development and knew how to turn coffee into theorems. The name recalls the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that converts coffee into theorems”.
Editing and coordination: Agate A. Timón G Longoria (ICMAT).
you can follow MATTER on Facebook, Twitter and Instagram, or sign up here to receive our weekly newsletter.