T-ERC_COG - Tremplin-ERC Consolidator Grant

Random Trees and Planar Metrics – RanTanPlan

Submission summary

(ERC abstract): The main purpose of this proposal is to explore random fractal 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 by extending the Brownian geometry to the stable (when the metric space has "holes" or "hubs") or causal (when a time dimension is priviledged) paradigms and studying other well- known models such as the Poisson line ensembles of Aldous-Kendall or models (possibly in high genus) coming from 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 expertise of several members of the Laboratoire de Mathematiques d'Orsay, and on the unique environment of University Paris-Saclay and neighboring institutions.

Project coordination

Nicolas CURIEN (Laboratoire de mathématiques d'Orsay)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

LMO Laboratoire de mathématiques d'Orsay

Help of the ANR 112,435 euros
Beginning and duration of the scientific project: May 2022 - 24 Months

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