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ORIGINAL FRENCH ARTICLE: Hasard? Vous avez dit Hasard?

by Alice Guionnet

Translated Friday 14 September 2018, by

To control the unknown by "measuring" chance, scientists have developed mathematical objects, random matrices. Alice Guionnet explains their genesis, and shows how this major discovery influences other sciences and many areas of society.

"I am often asked if there are still things to find in mathematics. The truth is, there are more and more things to look for. Random matrices, that are found everywhere, are a perfect example."

How dare we talk about the laws of chance? Isn’t chance the antithesis of any law? (...) Probability is opposed to certainty; therefore, this is what we do not know and therefore it seems that we cannot calculate it." This is how the mathématician Poincaré began his famous text on chance. However, the calculation of probabilities has invaded many sciences and society, particularly through statistics. And I and my colleagues spend our days doing these calculations!

Photo: Frederic Bella

How does it work? It all began in the 16th century with the work of the Italian Cardano. Cardano was a scientist and a doctor. He invented the cardan shaft, a mechanical part still used in automobiles. But he was also a great mathematician. He was the first to have the idea that chance can be measured. History says that Cardano was making ends meet by gambling and, in this spirit, he tried to understand what his chances were to make a double six. Even if the movement of a die is governed by known physical laws, a throw depends on too many parameters (initial position, throw direction, etc.) for the result to be predicted accurately. But the chance - or probability - of getting a face on top to be the six *can* be calculated. Cardano noted that all faces must have the same probability of appearing, so there is a one-in-six chance for each value. When he rolls two dice, the side on which one die falls must not influence the side on which the other die will fall: these events are *independent*! There must therefore be a 1/36 probability of performing a double six.

**The Law of Large Numbers**

These findings may now seem quite obvious, but they are at the heart of the basis for calculating probabilities. They make it possible to measure what is unknown, incalculable, by assigning to it a *probability*. And this makes it possible to predict, with a certain probability, what can happen. From that moment on, the theory of probability developed rapidly. One of the first questions was to be able to check that we had not made a mistake by assigning a probability of 1/6 to each face to appear in the roll of a die.

Let’s look at a simpler case: a coin. A priori both sides have the same chance to appear: 1/2. But how can we verify this? Common sense tells us that if we throw our coin 1,000 times, the "tail" (as in "heads or tails") should appear about 500 times. But beginning with what size "error" should we suspect that the coin is "fake" or that our mathematical model in incorrect?

The answer will come from Bernoulli and de Moivre in the 18th century. They show that there is a more than a 65% chance that the "tail" side will appear between 460 and

540 times, but a probability of less than 2% that it will occur less than 400 times. The latter case is very "unlikely" and, if we observe it, we will wonder if our model is bad. Maybe our coin is biased, and instead has a probability of 0.4 that it falls "tails"? We will be able to validate this new hypothesis by performing new throws. Bernoulli’s theorem is called the law of large numbers: it shows that if you throw a coin many times, the "heads" side should appear about half the time. De Moivre quantifies the error that can be made in relation to this average: it is of the order of the square root of the number of throws (here about 33) and is described by the "bell-shaped curve". It is this curve that tells us that we have a probability of less than 2% that, in 1000 flips of the coin, there will be less than 400 "tails".

**1,05 boys for one girl**

Laplace soon shows that these theorems are not limited to coins, but apply in many other circumstances, as long as they involve independent events. This allows him to use these theorems to answer questions from everyday life, such as in demographics. For example, it studies the number of boys in relation to the number of girls in different regions of France. In most cases, the average is similar: there are about 1.05 boys to 1 girl. The error in relation to this average is in the order of the root of the number of people counted, so the bell curve reappears! But what happened to make some regions have a different ratio? In any case, we should search for the cause of this phenomenon, which is no longer linked to chance! Is what way is this game rigged?

The bell curve is also used to carry out surveys, for example to determine the size of the sample to be surveyed in order provide a result within a permissible margin of error. As you may imagine, chance becomes a tote bag into which you can put many variables for which you have only inexact values. The bell curve has invaded many sciences and the objects of probabilistic mathematics have multiplied. For example, in physics, statistical mechanics seeks to explain the behaviour of matter by modelling the interactions of small particles that make up that matter. On this scale, many phenomena are unknown to us or are difficult to calculate: we therefore model them by random particles. Since there are many particles per gram of material, we expect, as in the law of large numbers, to see the emergence of an average behaviour that is no longer random. We can then verify that our models are good if this average behaviour is consistent with our observations. We can thus understand the phenomenon of magnetization of iron by describing it via the small magnetic moments that compose it. This approach extends to the living: we model the brain by neural networks. These networks are now at the heart of artificial intelligence.

**Determine the correlations**

The probabilistic object that interests me appeared in the early 20th century, in the work of the statistician Wishart. Wishart considers a large table of data and asks if they are

correlated. For example, let’s imagine that the table represents the performances of many athletes in different sports. How do we know if these results are correlated, and, for example, that if you’re good at football you should also be good in swimming. If the performance of all athletes in these two sports is proportional, we will conclude quite quickly that these results are related.... But how can we find more refined criteria, and

how can we know if what one observes is related at random, which might mean that these data are unrepresentative, or might they point to the existence of some possible biological reason?

**Information theory, and atomic physics**

Thus Wishart began to use large tables whose coefficients were randomly selected: such a matrix is called a "large random matrix". As in the coin toss game, the idea is to see if the data studied have the same properties as the data found in random matrices, and otherwise to look for additional assumptions that may have been made in generating our data. To do this, mathematicians study all kinds of random matrices, in order to understand their typical properties.

Even today, random matrices are still used to analyze data. With the advent of "big data", these tables are becoming larger and larger, and since mathematicians have fear of nothing, they assume that the size of these matrices tends to infinity!

But the random matrices did not remain limited to data tables. They have been become widespread in atomic physics to describe the physics of heavy nuclei. Even if these are described by quantum mechanics, one ends up with equations so complicated that we don’t know to solve them. In the 1950s, the physicist Wigner proposed a more simple method of analysis, based on random matrices. He found that the levels of the predicted energies were close to those observed! When this was not the case, Dyson proposed to search for physical properties that had not been taken into account in the modeling, in order to arrive at a model closer to reality.

Random matrices have appeared in many other sciences, such as in computing, with data compression, or with under-determined data problems such as Netflix, which recommends films to its customers based on a very small number of "like"s. I myself came to random matrices through neural networks. In short, as the magazine *Pour la science* recently titled it, "random matrices are everywhere!"

**New horizons**

But random matrices are first of all magnificent objects that have opened a new field of mathematics. They are involved not only in probability and analysis, but also in other areas of mathematics such as number theory or operator algebras. As mathematicians, we seek to understand the fundamental properties of these objects: what are their typical properties? What is the probability that they will behave differently?

When I say that I am a mathematician, I am often asked if there are still things to be found in mathematics. The truth is, there are more and more things to look for. On the one hand, the development of science raises new mathematical questions every day. On the other hand, the advancement of our knowledge opens up new horizons for us. Random matrices are a perfect example: unthinkable without the very concept of probability that finally appeared very late, their study is developing rapidly today, both through their applications, but also through the richness of the theory they generate. »

[The Academy of Sciences website:] "For science", No. 487, May 2018 : "The enigma of the random matrices. Why are these mathematical objects to be found everywhere?" [for sale here].

*« Vieillissement : un mouvement perpétuel ?* »,("Aging: a perpetual movement?), Alice Guionnet’s lecture to high school students at the Academy of Sciences, on the occasion of Mathematics Week, [March 13, 2018:]

*La plus grande valeur propre de matrices de covariance empirique*, ("The greatest eigenvalue for matrices of empurical covariance") by Sandrine Péché, 2006, online at:, the CNRS website dedicated to the popularization of mathematics.

« An Introduction to Random Matrices », by Greg W. Anderson, Alice Guionnet and Ofer Zeitouni, Cambridge University Press, 2010. on line

"Common sense tells us that if we throw a thousand times a coin, "tails" should come up about 500 times. But from which point on will we suspect a mistake that the coin is fake?»

**Profile**

Alice Guionnet, mathematician, member of the Academy of Sciences, is a research director at the CNRS in the unit of pure and applied mathematics at ENS Lyon. Known for her work on probabilities and large random matrices, she is notably holder of the CNRS silver medal and the Loève prize.