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

TL;DR

This paper establishes large deviation principles for Dyson Brownian motion on the circle and multiradial SLE, revealing their asymptotic behaviors and connecting to integrable systems and Loewner energy.

Contribution

It provides the first finite-time LDP for Dyson-type diffusions on the circle and characterizes the large-time behavior, also deriving an LDP for multiradial SLE as  .

Findings

Finite-time LDP for Dyson Brownian motion on the circle.

Large-time behavior characterized for Calogero-Moser-Sutherland systems.

LDP in Hausdorff metric for multiradial SLE as  .

Abstract

We show a finite-time large deviation principle (LDP) for "Dyson type" diffusion processes, including Dyson Brownian motion on the circle, for a fixed number of particles as the coupling parameter $\beta=8/\kappa$ tends to $\infty$. We also characterize the large-time behavior of finite-energy and zero-energy systems. Interestingly, the latter correspond to the Calogero-Moser-Sutherland integrable system. We use these results to derive an LDP in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$_\kappa$, as $\kappa \to 0$, with good rate function being the multiradial Loewner energy. Here, the main difficulty is that the curves have a common target point, preventing the configurational (global) approach. Our proof thus requires topological results in Loewner theory: 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 possibly at their common endpoint.