6 problems

Consider a random band-matrix setting with band width $W$ and coupling parameter $\lambda$, and let delocalization denote the regime in which eigenvectors are not localized. Existi…

Let $d$ be the spatial dimension, let $W$ be the band width, and let $C>0$ be a sufficiently large constant. A random band matrix is called delocalized in the sense used by the sou…

Let $g_{ij}$ denote the Gaussian entries of a Wigner matrix, and consider replacing their distribution by any subexponential distribution. Delocalization means that the mass of eac…

The F-model is a one-parameter sub-family of the $6$-vertex model indexed by $c>0$. It weights a homomorphism height function $f$ by $c^{-N(f)}$, where $N(f)$ is the number of diag…

Let $U:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $U(x)=U(-x)$ and … for some $\alpha>0$. For each $M>0$, define the interaction sequence $u^M=(u^M_n)$ by $u^M_n:=U(n/M)$. For…

Let $G(n,p)$ be an Erdős–Rényi random graph. Assume … for some constant $\epsilon>0$. Let $v$ be a random unit vector uniformly distributed on the $(n-1)$-dimensional unit sphere,…