Multi-time Loewner energy: rate function for large deviation

TL;DR

This paper explores the large deviation principle for certain Schramm-Loewner Evolution (SLE) processes, linking it to the multi-time Loewner energy, and applies these results to Dyson Brownian motion and boundary perturbations.

Contribution

It introduces the large deviation principle for half-watermelon and multi-radial SLE, connecting it to the multi-time Loewner energy, and offers new proofs and derivations for related stochastic processes.

Findings

Large deviation principle established for half-watermelon and multi-radial SLE.

Rate function identified as the multi-time Loewner energy.

New proof of large deviation for Dyson Brownian motion and boundary perturbation property.

Abstract

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.