Emergent Branching Structures in Random Geometry

1) Wider research context
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 is now part of a blossoming mathematical theory. My project is part of the geometrical approach of this study, broadly referred to as random geometry, from the. This line of research has seen a remarkable number of breakthroughs in the last two decades, from all directions, be they in the discrete (decorated planar maps, random permutations,...) or in the continuum (Brownian map, Liouville quantum gravity, SLE).

2) Objectives
My proposal hinges upon two intertwined work packages, namely Random discrete geometry and Liouville quantum gravity. In a nutshell, the first work package focuses on the large scale features of decorated planar maps (O(n), FK), and of random permutations which are ultimately connected to random geometry. In the second one, I will address some open questions related to SLE explorations of quantum surfaces, and random permutons which are related to the question of directed Liouville quantum gravity.

3) Methods
My project aims at developing powerful tools to explore new horizons in different 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 recent major breakthroughs in the fields of planar maps and random conformal geometry, coupled to statistical mechanics or SLE. I have developed in past research a new arsenal of techniques from branching structures (fragmentation, growth-fragmentation, etc.) which seem particularly sharp, as I used them in a variety of contexts, from the description of SLE explorations of the critical quantum disc to large increasing sequences of permutations sampled from the Brownian separable permuton.

4) Levels of innovation
The project addresses major current challenges in random geometry and statistical physics. Some of the problems mentioned in the proposal would lead the very first results for some model of planar maps (O(2), FK), permutations/permutons (polynomial behaviour of the longest increasing sequence), or directed Liouville quantum gravity. This set of problems forms, in my opinion, an innovative and coherent project, from an innovative point of view which aims at using new powerful ideas from branching to address random geometry questions. The project also balances between accessible and “riskier” projects, for which at least some interesting intermediate results seem reachable.

5) Primary researchers involved
William Da Silva (PI) and Nathanaël Berestycki (mentor).