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

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 $\beta =8\mathrm{/}\kappa$ tends to $\mathrm{\infty }$. 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.