Liouville Quantum Gravity and Conformal Probability

Liouville quantum gravity is a model for producing a random two-dimensional surface. The model was introduced by Polyakov 30 years ago in the context of string theory and is now a centerpiece of a branch of physics called conformal field theory. Liouville quantum gravity is a continuum theory, but it has discrete analogs. Just as there are discrete ways to create random paths (by drawing them on grids) there are also discrete ways to produce random surfaces (by gluing together identical unit squares along their edges). The PI will study both discrete and continuum random surfaces: their relationship to each other, their relationship to gauge theories and other aspects of theoretical physics, and their relationship to conformal random geometry and random curves such as the Schramm-Loewner evolution.

In particular, the aim is to show that, in some sense, if one builds a discrete random surface out of a large number of very small squares, it looks very much like the continuum random surfaces studied in Liouville quantum gravity. Moreover, when additional statistical physical structure (Ising or Potts models) is associated with the discrete models, this structure corresponds to continuum random geometric objects (conformal loop ensemble) on the continuum random surfaces.