Abstract:
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. The first half of my talk is an overview of these aspects. The second half of the talk is on the random annulus. Although the geometry locally looks the same as on simply connected surfaces, the conformal structure of a random annulus is now a random variable, since annuli with different moduli are not conformally equivalent. The law of the modulus for a uniformly sampled random annulus was predicted in string theory and quantum gravity. I will report the recent verification of this conjecture joint with Morris Ang and Guillaume Remy. Our method also yields several exact formulae on SLE that were predicted by Cardy (2006) via the non-rigorous Coulomb gas 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.