Conveners
Lecture 1: Tilings, Vertex Models and other remarkable combinatorial objects
- Philippe Di Francesco
Description
The study of Random tilings was initiated and first developed in physics, from the Dimer models to quasi-crystals, before it became a pure mathematics subject. On the other hand, Vertex models of statistical physics describe the long range effect of local interactions in crystals. In this lecture, we start from a remarkable coincidence between the number of configurations of the so-called six vertex model, describing an ideal square lattice crystal of ice in two dimensions and a rhombus tiling model of a suitable hexagon. This is a small part of an extraordinary sequence of combinatorial coincidences, none of which is yet understood in terms of natural bijections. We show how the integrability of the six-vertex model allows to count its weighted configurations via a determinant. The computation of this determinant and some of its limits allows to identify the counting with that of so-called Descending Plane Partitions, in bijection with some specific rhombus tiling problem. We then generalize the problem to triangular lattice ice, in the form of the twenty vertex model, and identify a suitable non-bijective domino tilings counting problem. Again, integrability allows to compute the numbers of configurations exactly, and to prove new coincidences, unsupported by natural bijections.
