I will present several sharp results on the path regularity of SLE (Schramm-Löwner evolution), a random fractal curve that is prominent in random conformal geometry and statistical physics. In particular, I will discuss the modulus of continuity, law of iterated logarithm, and variation regularity. Previous works have determined the optimal Hölder and p-variation exponent; our results improves them by providing the optimal gauge function with the correct logarithmic correction. As a key step in the proof, we obtain sharp estimates on the lower tail of the Minkowski content.
This talk is based on a joint project with Nina Holden.