A large deviation principle for the Schramm-Loewner evolution in the uniform topology

Abstract

We establish a large deviation principle for chordal SLE${}_{\kappa }$ parametrized by capacity, as the parameter $\kappa \to 0+$, in the topology generated by uniform convergence on compact intervals of the positive real line. The rate function is shown to equal the Loewner energy of the curve. This strengthens the recent result of E. Peltola and Y. Wang who obtained the analogous statement using the Hausdorff topology.