Hyperbolic surfaces and large random maps

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.

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

We also plan on studying random maps in high genus and to connect to models of 2-dimensional hyperbolic geometry such as the Brook–Makover model, random pants decompositions or Weil–Petersson random surfaces.We believe that the tools developed in the context of random planar maps, such as the systematic use of the spatial Markov property, the utilization of random trees to decompose and explore the surfaces, or the fine study of geodesic coalescence can be successfully applied to the aforementioned models.

We expect spectacular results and we 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, and theoretical physics, as well as mathematicians coming from other areas, in particular from combinatorics.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.