Loewner Dynamics for the Multiple SLE(0) Process
Recently Peltola and Wang introduced the multiple SLE(0) process as the deterministic limit of the random multiple SLE(kappa) curves as kappa goes to zero. 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).