skip to content
 

Almost sure multifractal spectrum of SLE

Presented by: 
E Gwynne Massachusetts Institute of Technology
Date: 
Wednesday 28th January 2015 - 10:00 to 11:00
Venue: 
INI Seminar Room 1
Abstract: 
Co-authors: Jason Miller (Massachusetts Institute of Technology), Xin Sun (Massachusetts Institute of Technology)

Suppose that $\eta$ is an SLE$_\kappa$ in a smoothly bounded simply connected domain $D \subset \mathbb C$ and that $\phi$ is a conformal map from the unit disk $\mathbb D$ to a connected component of $D \setminus \eta([0,t])$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1) \rightarrow [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial \mathbb D$ such that $|\phi'( (1-\epsilon) x)| = \epsilon^{-s+o(1)}$ as $\epsilon \rightarrow 0$. I will present a rigorous computation of the a.s. multifractal spectrum of SLE (joint with J. Miller and X. Sun), which confirms a prediction due to Duplantier. The proof makes use of various couplings of SLE with the Gaussian free field. As a corollary, we also confirm a conjecture of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE.

The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons