Hyperbolic surfaces and large random maps

Introduction

The main purpose of this proposal is to explore random planar metrics. Two canonical models of random continuum surfaces have been introduced in the past decade, namely the Brownian sphere obtained as the scaling limit of uniform random planar triangulations, and the Liouville Quantum Gravity metric obtained formally from the exponential of the Gaussian free field on the sphere.

Objectives

Our objective is to broaden our understanding of random planar metrics to the case of metrics with “holes” or “hubs”, and to the causal paradigm (when a time dimension is singled out).

1. We also plan on studying random maps in high genus.
2. We aim to connect to models of 2-dimensional hyperbolic geometry such as:
   • The Brook–Makover model
   • Random pants decompositions
   • Weil–Petersson random surfaces

Methodology

We believe that the tools developed in the context of random planar maps can be successfully applied to the aforementioned models. These tools include:

• The systematic use of the spatial Markov property
• The utilization of random trees to decompose and explore the surfaces
• The fine study of geodesic coalescence

Expected Outcomes

We expect spectacular results and hope to reinforce the connections between those very active fields of mathematics. This proposal should give rise to exceptionally fruitful interactions between specialists of different domains such as:

• Probability theory
• Two-dimensional hyperbolic geometry
• Theoretical physics
• Mathematicians from other areas, particularly from combinatorics

Support and Environment

To ensure the best chances of success for the proposed research, we will rely on the unique environment of University Paris-Saclay and neighboring institutions.