We analyse the spectrum of additive finite-rank deformations of $N \times N$ Wigner matrices $H$. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue $d_i$ of the deformation crosses a critical value $\pm 1$. This transition happens on the scale $|d_i| - 1 \sim N^{-1/3}$. We allow the eigenvalues $d_i$ of the deformation to depend on $N$ under the condition $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$. We make no assumptions on the eigenvectors of the deformation. In the limit $N \to \infty$, we identify the law of the outliers and prove that the non-outliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the \emph{isotropic local semicircle law}, which establishes optimal high-probability bounds on the quantity $< v,[(H - z)^{-1} - m(z) 1] w >$, where $m(z)$ is the Stieltjes transform of Wigner's semicircle law and $v, w$ are arbitrary deterministic vectors.

We present a general computational framework to locally characterize any shape space of meshes implicitly prescribed by a collection of non-linear constraints. We computationally access such manifolds, typically of high dimension and co-dimension, through first and second order approximants, namely tangent spaces and quadratically parameterized osculant surfaces. Exploration and navigation of desirable subspaces of the shape space with regard to application specific quality measures are enabled using approximants that are intrinsic to the underlying manifold and directly computable in the parameter space of the osculant surface. We demonstrate our framework on shape spaces of planar quad (PQ) meshes, where each mesh face is constrained to be (nearly) planar, and circular meshes, where each face has a circumcircle. We evaluate our framework for navigation and design exploration on a variety of inputs, while keeping context specific properties such as fairness, proximity to a reference surface, etc.

We consider a general class of symmetric or Hermitian random band matrices $H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket^d}$ in any dimension $d\ge 1$, where the entries are independent, centered random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. We assume that $s_{xy}$ vanishes if $|x-y|$ exceeds the band width $W$, and we are interested in the mesoscopic scale with $1\ll W\ll N$. Define the {\it{generalized resolvent}} of $H$ as $G(H,Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix with entries $Z_{xx}\in \mathbb C_+$ for all $x$. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions $d\ge 2$. More precisely, for any fixed $d\ge 2$, we prove that the bulk eigenvectors of $H$ are delocalized in certain averaged sense if $N\le W^{1+\frac{d}{2}}$. This improves the corresponding results in \cite{HeMa2018} under the assumption $N\ll W^{1+\frac{d}{d+1}}$, and in \cite{ErdKno2013,ErdKno2011} under the assumption $N\ll W^{1+\frac{d}{6}}$. For 1D random band matrices, our fluctuation averaging result was used in \cite{PartII,PartI} to prove the delocalization conjecture and bulk universality for random band matrices with $N\ll W^{4/3}$.

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