Emergent Branching Structures in Random Geometry

The pionneering work of the physicist Polyakov, more than forty years ago, opened the way for a new approach in string theory based on the study of random surfaces, which are fundamental and universal mathematical objects. This project aims at developing powerful tools to explore new horizons in several aspects of random geometry. The main objective is to show how emergent branching structures cast a light on the large scale limit of certain features of statistical physics models at criticality. The project builds on a remarkable number of recent major breakthroughs in the fields of planar maps and random conformal geometry, coupled to statistical mechanics or Schramm-Loewner evolution. The project hinges upon two intertwined aspects of random geometry, namely random discrete geometry and Liouville quantum gravity. The first aspect focuses on the large scale features of decorated planar maps (loop-O(n), Fortuin- Kasteleyn), and of random permutations which are ultimately connected to random geometry. In the second one, we investigate some open questions related to Schramm-Loewner type of explorations on quantum surfaces, and random permutons which show up in the limit of natural models of large random permutations.