L'invariance conforme et l'universalité au point critique des modèles bidimensionnels

Des quelques articles publiés par Robert P. Langlands en physique mathématique, c’est celui publié dans le Bulletin of the American Mathematical Society sous le titre Conformal invariance in two-dimensional percolation qui a eu, à ce jour, le plus d’impact : les idées d’Oded Schramm ayant mené à l’équation de Loewner stochastique et les preuves de l’invariance conforme de modèles de physique statistique par Stanislav Smirnov ont été suscitées, au moins en partie, par cet article. Ce chapitre rappelle sommairement quelques idées de l’article original ainsi que celles issues des travaux de Schramm et Smirnov. Il est aussi l’occasion pour moi de décrire la naissance de ma collaboration avec Robert Langlands et d’exprimer ma profonde gratitude pour cette fantastique expérience scientifique et humaine.Of all mathematical physics contributions by Robert P. Langlands, the paper Conformal invariance in two-dimensional percolation published in the Bulletin of the American Mathematical Society is the one that has had, up to now, the most significant impact : Oded Schramm’s ideas leading to the stochastic Loewner equation and Stanislav Smirnov’s proof of the conformal invariance of percolation and the Ising model in two dimensions were at least partially inspired by it. This chapter reviews briefly some ideas of the original paper and some of those by Schramm and Smirnov. This chapter is also for me the occasion to reminisce about the extraordinary scientific and human experience that working with Robert Langlands was. It started in the late 80’s when Langlands would spend Summers at the Centre de recherches mathématiques in Montreal. The “Langlands program” was already launched and many colleagues were devoting their career to it. Beside his steady efforts in automorphic forms, Langlands was already exploring new fields, mathematical physics being one of them. He studied conformal field theory, just then introduced, and started thinking about the renormalisation group. He presented some of these ideas in a study workshop in Montreal and this is when our collaboration took off. This collaboration concentrated on problems related to conjectures of universality and conformal invariance of two-dimensional discrete systems on compact domains, and on the Bethe Ansatz. Discussing, bouncing ideas and simply collaborating with Langlands was a fantastic experience. I had a hard time understanding his more formal presentations. But one-on-one discussions at the blackboard were always concrete, instructive and fruitful. My barrage of questions never seemed to frazzle him. Whenever he understood where I was blocked, his answer would often be “Let me give you an example”. I had imagined that he would prefer the loftier way of mathematical communication through abstraction. But it was a nice surprise to discover that he knew so many concrete examples that revealed the crux of difficult mathematical concepts. I am deeply indebted to him for this collaboration that lasted about ten years and for his friendship that remains very much alive today

The Jordan Structure of Two Dimensional Loop Models

We show how to use the link representation of the transfer matrix $D_N$ of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin-Kasteleyn model with various boundary conditions and parameter $\beta = 2 \cos(\pi(1-a/b)), a,b\in \mathbb N$ and, more specifically, partition functions of the corresponding $Q$-Potts spin models, with $Q=\beta^2$. The braid limit of $D_N$ is shown to be a central element $F_N(\beta)$ of the Temperley-Lieb algebra $TL_N(\beta)$, its eigenvalues are determined and, for generic $\beta$, a basis of its eigenvectors is constructed using the Wenzl-Jones projector. To any element of this basis is associated a number of defects $d$, $0\le d\le N$, and the basis vectors with the same $d$ span a sector. Because components of these eigenvectors are singular when $b \in \mathbb{Z}^*$ and $a \in 2 \mathbb{Z} + 1$, the link representations of $F_N$ and $D_N$ are shown to have Jordan blocks between sectors $d$ and $d'$ when $d-d' < 2b$ and $(d+d')/2 \equiv b-1 \ \textrm{mod} \ 2b$ ($d>d'$). When $a$ and $b$ do not satisfy the previous constraint, $D_N$ is diagonalizable.Comment: 55 page

Geometric Exponents of Dilute Logarithmic Minimal Models

The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is beta = -2cos(pi/barkappa}) where the parameter barkappa is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges c = 13 - 6(barkappa + barkappa^{-1}) and, for the minimal models of interest here, barkappa = p/p' where p and p' are two coprime integers. The geometric exponents of the hull and external perimeter are studied for the pairs (p,p') = (1,1), (2,3), (3,4), (4,5), (5,6), (5,7), and that of the red bonds for (p,p') = (3,4). Monte Carlo upgrades are proposed for these models as well as several techniques to improve their speeds. The measured fractal dimensions are obtained by extrapolation on the lattice size H,V -> infinity. The extrapolating curves have large slopes; despite these, the measured dimensions coincide with theoretical predictions up to three or four digits. In some cases, the theoretical values lie slightly outside the confidence intervals; explanations of these small discrepancies are proposed.Comment: 41 pages, 32 figures, added reference