Multi-time Loewner energy: rate function for large deviation

The classification of probability measures that satisfy both conformal invariance and domain Markov property is equivalent to characterizing solutions to the Belavin--Polyakov--Zamolodchikov (BPZ) equations, as established by Dub\'edat~[Dub07]. In this context, the partition functions for half-watermelon SLE and for multi-radial SLE serve as fundamental solutions to the BPZ equations. In this article, we investigate the large deviation principle for both half-watermelon SLE and multi-radial SLE. The associated rate function is given by the multi-time Loewner energy, introduced in~[CHPW26]. As applications, we provide an alternative proof of the large deviation principle for Dyson Brownian motion, as well as a new derivation of the boundary perturbation property of the multi-time Loewner energy.