Random sphere, disk, and annulus

The random geometry on simply connected surfaces is a well established subject in probability. The key aspects of this theory include the scaling limit of random planar maps, Liouville quantum gravity, Schramm–Loewner evolution (SLE), and Liouville conformal field theory. For non-simply connected surfaces such as a random annulus, a new feature arises. Since annuli with different moduli are not conformally equivalent, the modulus of a random annulus is now a random variable. The law of the modulus for a uniformly sampled random annulus was predicted in string theory and quantum gravity. I will report the recent proof of this conjecture jointly with Morris Ang and Guillaume Remy. Time permitting, I will explain how our method can be applied to obtain several exact formulae on SLE that were predicted in physics via the non-rigorous method, including the generating function of the number of non-contracting loops for a conformal loop ensemble on the annulus, and the annulus partition function for Werner's self-avoiding loop.