Percolation
Author's comments: Problems of endoscopy first arose as I began the study of Shimura varieties in Bonn during the academic year 1970/71. I reflected on them for a long time, in part in collaboration with Labesse, in part in collaboration with Shelstad. I presented a fairly mature form of my reflections in the Paris lectures, Les débuts d'une formule des traces stable, in which the presence of a major obstacle, overcome considerably later through the efforts of a number of mathematicians, in particular Waldspurger and Ngô, was clearly described. It was labelled the fundamental lemma. I continued trying to find a proof during the eighties, but was no doubt discouraged and searching for a diversion.
I turned, at some point, although not completely, away from automorphic forms to mathematical questions of an altogether different kind, about matters that I had not considered since I was an undergraduate, studying, as I recall, among others the book of Lanczos on The variational principles of mechanics and the book of Whitham on Linear and nonlinear waves, but also less classical aspects of mathematics physics, among them the biography of Einstein, Subtle is the Lord... by Pais, which is of course a serious introduction to modern physics.
By good luck, I fell into conversation one afternoon with the physicist Giovanni Gallavotti, who was visiting the School of Mathematics at the Institute for the academic year 1984-85 as a participant in a special program in mathematical physics that I had, I recall, suggested and encouraged, but only as a spectator. He explained to me, as we strolled through the meadow that surrounded the Institute, the problem of renormalization. I was, in spite of my ignorance, fascinated by it and tried to learn more. Renormalization is, of course, a term that I had seen before in connection with quantum field theory, but that was an area with which I had no familiarity. I learned from Gallavotti that it was, in particular, associated to dynamical systems in an infinite number of variables with a fixed point that had only a finite number of expanding directions and in which the eigenvalues of the linearized transformation, all but a finite number less than \(1\) in absolute value, tended to \(0\). So for me, a mathematician and not a mathematical physicist, and certainly not a physicist, the immediate problem became the construction of such systems together with a proof that they possessed this property.
It was not of course that no examples were available. What was missing was a clear mathematical definition of systems of this type, of their fixed points, and proofs of the desired structure. Since we are dealing, at first, with infinite-dimensional spaces with no precise definition, thus with no obvious coordinate system and no exact notion of admissible points, some reflection about the basic notions was---and remains---necessary. The simplest example seemed, after some time spent with the pertinent literature, to be percolation. I bought and studied with some care the book of Kesten, Percolation theory for mathematicians, which had appeared not long before, in 1982. The possibility slowly occurred to me that the crossing probabilities studied with considerable success by Kesten might yield the desired fixed point. Some care has to be taken with this statement, because there could be---and is---a continuous family of fixed points. In the present context they are defined by symmetries that manifest themselves in the form of conformal invariance, to which I shall return.
My first experiment with this possibility, which had, so far as I know, not been earlier examined in the literature, was to construct a finite-dimensional model of percolation in the paper Finite models for percolation, in which the presentation has, thanks to my co-author Marc-André Lafortune, at the time an undergraduate at the Université de Montréal and much more practiced with computers than I, considerable elegance.
Later on, in a discussion with Yvan Saint-Aubin, a colleague at the same university, I explained my developing views on the crossing probabilities, thus that they could serve as the coordinates of the fixed point. This, correctly interpreted, means that the crossing probabilities are universal. They are not absolutely universal, only universal for systems subject to certain constraints, in particular to the same constraints on symmetry, for example translational invariance and an appropriate reflection-symmetry. Saint-Aubin's first reaction was sceptical, a justified scepticism. So we decided to examine the question numerically. The strategy to be used was, for various reasons, largely in the hands of Saint-Aubin, who had had much more experience with computers than I. We were joined by a student and by a colleague. The experiments and their results are clearly stated in the abstract to the paper On the universality of crossing probabilities in two-dimensional percolation. I give it here because it was not reproduced with the paper.
`Six percolation models in two dimensions are studied: percolation by sites and by bonds on square, hexagonal, and triangular lattices. Rectangles of width \(a\) and height \(b\) are superimposed on the lattices and four functions, representing the probabilities of certain crossings from one interval to another on the sides, are measured numerically as functions of the ratio \(a/b\). In the limits set by the sample size and by the conventions and on the range of the ratio \(a/b\) measured, the four functions coincide for the six models. We conclude that the values of the four functions can be used as coordinates of the renormalization-group fixed point.´
The models are chosen so that the crossing probabilities in all models would have considerable symmetry, indeed as it turned out a rotational symmetry and, of course, translational symmetry. Symmetry was, however, not our main concern. That was universality. However, I had already as we prepared our results for publication discussed them with Michael Aizenman and Thomas Spencer. Aizenman then suggested the possibility of conformal invariance, which we began to test immediately. The conclusions are presented in a subsequent paper, Conformal invariance for two-dimensional percolation . This second paper had a more immediate influence than the first, above all on Oded Schramm, who unfortunately lost his life not many years later in a climbing accident. As I remember the one conversation I had with him, he mentioned that this paper played a role in his creation of the theory of Schramm-Loewner Evolution.
Both papers are mentioned, but only incidentally, in the laudatios for the two Fields medals related to conformal invariance and percolation: the one by Charles Newman for Wendelin Werner and the one by Harry Kesten for Stanislaw Smirnov. The one by Kesten is misleading and misleading in an important, although presumably unintended, way. Kesten refers to a sentence in the first paper, "Conversations with Michael Aizenman have greatly clarified our views as to the nature of the universality manifested by the crossing probabilities, and our understanding of their invariance under various transformations of the curves defining the event E. In particular, they have suggested a number of conjectures to which we shall return in a later paper, in which the modifications required for models with less symmetry than those treated here will also be discussed." In the second, the statement "Conversations with Michael Aizenman after the data were in hand greatly clarified for us their nature. In particular he suggested that these crossing probabilities would be conformally invariant." Kesten, however, refers to the first sentence alone and deduces from it the following conclusion, taken from his laudatio, "... we did not specify what it means that the scaling limit exists and is conformally invariant. It seems that M. Aizenman (see [13], bottom of p. 556) was the first to express this as a requirement about the scaling limit of crossing probabilities." The reference is to the first paper. In other words, Kesten confounds the problem of conformal invariance with that of the existence of an essentially unique scaling limit and this confounds, in my view, two, even three, problems of quite different depth. The existence of the scaling limit had been proved by Kesten himself; the essential uniqueness is related to the very broad collection of problems raised by renormalization; conformal invariance will be, if my intuition is correct, relatively easy to deduce in general from a theory of renormalization and its fixed points, a theory that will necessarily include universality, if that can be established. The probabilists are very attached to conformal invariance and why not? The Scramm-Loewner theory is very elegant and the results of Smirnov significant. It is nevertheless renormalization that is the broader issue and deserved to be discussed in at least one of the two laudatios.
The view of critical phenomena revealed to me by the initial conversation with Gallavotti as associated to infinite-dimensional dynamical systems and their fixed-points was tremendously appealing and percolation appeared to be a promising place to begin. The questions themselves are, however, omnipresent in modern physics: in statistical physics, in quantum field theory, in fluid dynamics. It is, I found, difficult, even impossible, for a mathematician, at least this mathematician, to find his bearings in the variety of approximations, guesses, and intuitions, and insights available, some more, some less convincing. It is hard to know how or where to begin to think about all the questions that appear.
Even for percolation, a space to contain the fixed-points was needed, as well as possible transformations (the dynamical system) to distinguish the pertinent fixed points. So there is a transition to be made, from the lattices with probabilities of occupation to the dynamical system, in this instance, the passage to an object defined by crossings. Even with Kesten's theorem in hand it is not entirely evident what to do. The theorem takes us a long way, but the universality must still be proved. My intuition, which is not supported by much evidence, is that a proof of universality will inevitably contain in a more or less clear manner a proof of conformal invariance. It is because of this intuition, which I expect to be more widely applicable and valid not just for percolation, that I find it more than unfortunate that Kesten has confounded universality with conformal invariance.
The following section, the one on mathematical physics, contains little, perhaps nothing, of interest. I tried for some time to acquire a useable understanding of the various domains of physics and mathematical physics that would lead to some concrete definitions and questions that permitted, on the one hand, the formulation of real mathematical theorems or even conjectures and that, on the other, were relevant to the insights of the mathematical physicists or to an understanding of the natural phenomena of, say, fluid dynamics. I made a real effort and learned quite a bit, but not enough. I had always hoped to return to these questions, not with any hope of accomplishing something but just to educate myself. This appears less and less likely.