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Suppose that $\eta$ is a Schramm-Loewner evolution ($\SLE_\kappa$) in a smoothly bounded simply connected domain $D \subset \C$ and that $\phi$ is a conformal map from $\D$ to a connected component of $D \setminus \eta([0,t])$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1) \to [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial \D$ such that $|\phi'( (1-\epsilon) x)| = \epsilon^{-s+o(1)}$ as $\epsilon \to 0$. We rigorously compute the a.s.\ multifractal spectrum of $\SLE$, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the a.s.\ bulk integral means spectrum of $\SLE$, we obtain the optimal H\"older exponent for a conformal map which uniformizes the complement of an $\SLE$ curve, and we obtain a new derivation of the a.s.\ Hausdorff dimension of the $\SLE$ curve for $\kappa \leq 4$. Our results also hold for the $\SLE_\kappa(\ul\rho)$ processes with general vectors of weight $\ul\rho$.

The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multitime and multilocation distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large-time limit in the so-called relaxation time scale.

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For Komatu-Loewner equation on a standard slit domain, we randomize the Jordan arc in a manner similar to that of \cite{S} to find the SDEs satisfied by the induced motion $\xi(t)$ on $\partial\HH$ and the slit motion $\s(t)$. The diffusion coefficient $\alpha$ and drift coefficient $b$ of such SDEs are homogenous functions. Next with solutions of such SDEs, we study the corresponding stochastic Komatu-Loewner evolution, denoted as ${\rm SKLE}_{\alpha,b}$. We introduce a function $b_{\rm BMD}$ measuring the discrepancy of a standard slit domain from $\HH$ relative to BMD. We show that ${\rm SKLE}_{\sqrt{6},-b_{\rm BMD}}$ enjoys a locality property.