9 problems

Let $\rho$ be a probability density with moments of all orders, meaning that … for every $q\geq 1$. Let $E_o\in\mathbb R$. Suppose there is a $\delta>0$ such that, on … the symbol…

Let $G(V,E)$ be a connected infinite graph with bounded vertex degrees. Assume that $G$ is amenable and is an abstract quasicrystal graph: for every radius $r$ and every $r$-patter…

Let $d\in\{1,2,3\}$. For each $n\ge 1$, let $\varphi_{n,L}$ be the corresponding eigenfunction, let $U_{n,L}\in[-L/2,L/2]^d$ be a point where $|\varphi_{n,L}|$ reaches its global m…

Poisson point-process and Gumbel conjecture. The point process

Let $p$ be the Bernoulli parameter, let $\beta^{-1}(n)$ denote the decreasing sequence appearing in the model, and let $I_{p,\zeta}(x)$ be the integrated density of states at disor…

Let $T$ be the random weighted shift associated with independent weights $\{X_n\}_{n=1}^{\infty}$ on a Hilbert space $\mathcal{H}$, and let $\{e_k\}_{k=1}^{\infty}$ be its associat…

Bauer–Golinelli conjecture. The following phase transition occurs: if $c<e$, then $\mu_c$ has no extended states at $0$; if $c>e$, then $\mu_c$ has extended states at $0$.

Let $E$ be distributed according to the arcsine law, $U$ be uniform on $[0,1]$, and let $\mathcal{Z}$ be a standard two-sided Brownian motion started from $0$, independent of $E$ a…

Let ${\mathbf s}$ be a sequence and let $H_\lambda={\mathcal A}_{\mathbf s}+\lambda {\mathcal V}$ be the Anderson model on the antitree ${\mathbb A}_{\mathbf s}$, with disorder par…