We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\Z/L\Z)^d$. The entries $h_{xy}=\overline h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$, which satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any fixed $n\in \N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$, with an error $\OO(W^{-nd/2})$, and use it to prove a local law on the Green's function. This $T$-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions $d\ge 8$ in part I \cite{PartI_high} of this series.

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.