Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+)

Abstract

We show a finite-time large deviation principle (LDP) for "Dyson type" diffusion processes, including Dyson Brownian motion (DBM) on the circle, for a fixed number of particles as the coupling parameter $\beta=8/\kappa$ tends to $+\infty$. Zero-energy systems correspond to the Calogero-Moser-Sutherland integrable system. We also characterize the large-time behavior of finite-energy systems: zero-energy systems approach exponentially fast a static equally-spaced configuration, while finite-energy systems may have polynomial convergence rates, and the system may never become static. We use our DBM result to derive a finite-time LDP in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$_\kappa$, as $\kappa$ tends to $0+$, with good rate function being the multiradial Loewner energy. Using a derivative estimate for the radial Loewner map in terms of the energy of its driving function, we show that finite-energy multiradial Loewner hulls are disjoint unions of simple curves, except at their common endpoint.