Random geometry from probability theory to quantum gravity


Probability theory deals with the construction and analysis of random objects, like Brownian motion which describes a natural random continuous trajectory and the universal limit of random walks. Is there an equally universal notion of random geometry on a manifold? It has been answered in the affirmative in the 2D case, in the form of the Brownian sphere, but it is an open problem in higher dimensions. This mathematical question happens to be closely related to the problem of quantum gravity: under suitable physical constraints a random geometry on the spacetime manifold provides a realization of the Euclidean gravitational path integral. Indeed, the Brownian sphere is exactly the random metric associated to two-dimensional (Liouville) quantum gravity. I will discuss the challenges and prospects on route to random geometry in higher (and from the gravitational viewpoint more realistic) number of dimensions.

NH Koningshof, Veldhoven, Netherlands