A new random geometry by Jean-François Le Gall

Jean-François Le Gall gave a non-technical description of his work for the Paris Academy of Sciences. We give below a translation into English of the article Une nouvelle géométrie aléatoire.

A new random geometry

Before telling you about my own research in probability theory, I would like to begin with a few words about the history of this area of mathematics. The history of probability begins in the sixteenth and seventeenth centuries with the Italian mathematician Cardano, and especially the famous correspondence between Pascal and the Chevalier de Méré [Antoine Gombaud (1607-1684)]. The latter who, according to a letter from Pascal to Fermat, "had a very good mind but was not a geometer" posed to Pascal several problems relating to games of chance, including that of knowing whether one has more chances of obtaining a six by rolling a die 4 times, or getting a double six by rolling two dice 24 times. Pascal, and also Fermat, took the Chevalier de Méré's questions seriously and provided elegant solutions. Many famous mathematicians then became interested in what was then called the calculation of probabilities and established important results: the famous law of large numbers, which states in a particular case that the proportion of the number of occurrences of heads or tails in a long series of tosses is close to

\large\frac{1}{2}\normalsize

, was established by Jacob Bernoulli and generalised by others. Despite this progress, probability remained a sort of poor relation of mathematics (and was often assimilated to a part of physics) until the beginning of the twentieth century, due to a lack of rigorous axiomatic foundations.

Everything changed in 1933 when the great Russian mathematician Kolmogorov took advantage of the recent invention of measurement theory to base probabilities on a powerful axiomatic, which is used today by almost all specialists. Despite this, probability theory remained an object of suspicion for many mathematicians for a long time: the brilliant Paul Lévy, one of the biggest names in probability theory, was not elected to the mathematics section of the Academy of Sciences until he reached the ripe old age of 78. However, Kolmogorov's formalism today makes it possible to consider all kinds of mathematical objects depending on chance, with a very large number of applications. The best example is undoubtedly the mathematical Brownian motion, which is a curve depending on chance, accounting for the phenomenon studied by great physicists like Albert Einstein and Jean Perrin. To give an intuitive idea of what Brownian motion is, we imagine a walker moving in a very large park and taking a step every second in one of the four possible directions, North, South, East or West, chosen randomly every time. If we observe the movement of the walker over a long period of time, say a few hours, and on a suitable scale, we will see a random curve which is close to that of Brownian motion in dimension two. Brownian motion is thus a sort of ideal prototype of a purely random curve.

My first research work dealt with various properties of the Brownian motion curve, notably concerning the intersections of this curve with itself: for example, we show that the Brownian motion in the plane passes an infinite number of times at certain points, and my work has enabled us to better understand this phenomenon. I must here pay tribute to my former thesis director Marc Yor, who unfortunately left us last January. It was he who, through his infectious enthusiasm, was able to convince me to take an interest in the fascinating mathematical object that is Brownian motion.

My recent research focuses on random geometry: it is no longer a question of studying a curve depending on chance, like Brownian motion, but a whole geometry with a notion of distance between any two points in space. One of the motivations again comes from theoretical physics, and more particularly from what is called the theory of quantum gravity in two dimensions. How do we construct this random geometry? Let's imagine a sphere, for example the surface of our Earth where the oceans have disappeared, on which there are a large number of cities. These towns are linked together by roads, each town being connected to a small number (3 or 4 for example) of other towns. Two roads can only cross in one city, and it is always possible to go from one city to another by chaining a certain number of steps. If we give ourselves two cities A and B, we define the distance between A and B as being proportional to the minimum number of cities that we must cross to go from city A to city B - in a pictorial way, we can think that traveling on the roads is very fast, but crossing cities takes a long time. There is a precise mathematical process for choosing completely randomly the roads between cities, and we thus obtain a notion of random distance between any two cities. If the number of cities is very large, we can consider that any point on the sphere is very close to a city and we will obtain a random geometry on the sphere: this leads to the mathematical model called the Brownian map (map like a geographical map or road, and Brownian because of important links with Brownian motion) whose existence and uniqueness I have shown, with other researchers. This subject is a place for very fruitful meetings between probabilistic mathematicians, specialists in combinatorics and graph theory, and theoretical physicists. Like Brownian motion, the Brownian map is an extraordinary object, which I am sure will continue to fascinate mathematicians for a long time. Beyond mathematics, remembering that the non-Euclidean geometries discovered by Riemann allowed Einstein to invent the theory of relativity, we can dream a little and hope that these new random geometries will provide new keys to understanding the quantum world that surrounds us.

Last Updated March 2024