Large deviations of radial SLE$_{\infty }$

Abstract

We derive the large deviation principle for radial Schramm-Loewner evolution ($\operatorname {SLE}$) on the unit disk with parameter $\kappa \rightarrow \infty $. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures $\{\phi _{t}^{2} (\zeta )\, d\zeta \}_{t \in [0,1]}$ on the unit circle and equals $\int _{0}^{1} \int _{S^{1}} |\phi _{t}'|^{2}/2\,d\zeta \,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.