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

Abstract

We derive the large deviation principle for radial Schramm-Loewner evolution ($\mathrm{SLE}$) on the unit disk with parameter $\hat{I}º\to \infty$. 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 $\{\ddot{I}{•}_t^2(\hat{I}\P )d\hat{I}\P {\}}_{t\in [0,1]}$ on the unit circle and equals ${\int }_0^1{\int }_{S^1}|\ddot{I}{•}_t^{\prime }{|}^2/2d\hat{I}\P dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.

17 pages, 1 figure, revised according to referee's report