Loewner Dynamics for the Multiple SLE(0) Process

Abstract

Recently Peltola and Wang introduced the multiple SLE(0) process as the deterministic limit of the random multiple SLE($\kappa$) curves as $\kappa \to 0$. They prove this result by means of a small kappa large deviations principle, but the limiting curves also turn out to have important geometric characterizations that are independent of their relation to SLE($\kappa$). In particular, they show that the SLE(0) curves can be generated by a deterministic Loewner evolution driven by multiple points, and the vector field describing the evolution of these points must satisfy a particular system of algebraic equations. We show how to generate solutions to these algebraic equations in two ways: first in terms of the poles and critical points of an associated real rational function, and second via the well-known Caloger-Moser integrable system with particular initial velocities. Although our results are purely deterministic they are again motivated by taking limits of probabilistic constructions, which I will explain. Joint work with Nam-Gyu Kang (KIAS) and Nikolai Makarov (Caltech).