A combinatorial approach to random hyperbolic surfaces


Starting with dynamical triangulations of the string world sheet and matrix models, random maps have occupied a central place in the study of 2d (Euclidean) quantum gravity. Advances in combinatorics (e.g. tree bijections) and probability theory (e.g. Gromov-Hausdorff limits of random metric spaces) led to a rigorous construction of 2d quantum gravity in the form of Brownian geometry on surfaces, and its identification with Liouville Quantum Gravity. In this talk I will describe how some of these methods extend naturally to random hyperbolic geometry on surfaces, a natural alternative to random maps. In particular, I will show how a bijection between the moduli space of genus-0 hyperbolic surfaces (with boundaries) and certain labeled trees provides insight into the associated random metric space. Based on joint works with N. Curien and with T. Meeusen and B. Zonneveld.

IHP, Paris