  The main motivation of the present work is to investigate the asymptotic
behavior as $\kappa \to 0+$ of multiradial Schramm-Loewner evolution,
SLE$_\kappa$. We show that this process with the common parameterization
satisfies a finite-time large deviation principle (LDP) in the Hausdorff metric
with non-negative rate function, the multiradial Loewner energy. We also
characterize the large-time behavior of curves with finite energy and zero
energy (whose driving functions correspond to the trigonometric Calogero-Moser
system).
  The first half of this article is of independent interest regardless of SLE
theory. It is devoted to proving a finite-time LDP for Dyson Brownian motion on
the circle for a fixed number $n$ of particles as the coupling parameter
$\beta=8/\kappa$ tends to $\infty$. To our knowledge, in the literature large
deviations of Dyson Brownian motion has only been considered for fixed $\beta$
and as $n$ tends to $\infty$. While the non-Lipschitz drift precludes the
application of the Freidlin-Wentzell theorem, we show that the rate function
has the same form as in Freidlin-Wentzell theory for diffusions with uniformly
Lipschitz drift.
  In the second half of this article, we turn to proving an LDP for multiradial
SLE$_\kappa$. Here, the main technical difficulty is that the SLE$_\kappa$
curves have a common target point, preventing the usual configurational, or
global, approach. Instead, we make careful use of the contraction principle
from the LDP for Dyson Brownian motion (proven in the first part of the
article), combined with topological results in Loewner theory: we show that
finite-energy multiradial Loewner hulls are always disjoint unions of simple
curves, except possibly at their common endpoint. A key to this is obtained
from a derivative estimate for the radial Loewner map in terms of the energy of
its driving function.
