LA Probability Forum

Schramm-Loewner evolution SLE(k) is a one-parameter family of random curves arising from two-dimensional conformal geometry, which in the limit k->0 converge to deterministic curves (hyperbolic geodesics). In such a large deviations limit, a precise large deviation principle (LDP) has been proven in some cases involving interacting SLE curves. The rate function, termed Loewner energy, plays an important role for the LDP, and also provides rich interplay with algebraic geometry and Teichmüller theory. In this talk, we shall focus on the probabilistic aspects of the LDP for SLE curves and their Loewner driving processes, which are variants of Brownian motion. Specifically, we discuss the chordal, radial, multichordal, and multiradial cases, and the explicit relationship of the latter to Dyson Brownian motion. In particular, we consider an LDP for Dyson Brownian motion, which may also be of independent interest in probability theory and random matrix theory.

Based on joint works with O. Abuzaid, V. Healey, and Y. Wang.