Conformal invariance and universality at the critical point of two-dimensional models

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 1980s when Langlands would spend Summers at the Centre de recherches mathematiques 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 renormalization 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 10 years and for his friendship that remains very much alive today.