55 problems

Let the tridiagonal random matrix models of Dumitriu and Edelman be scaled appropriately so that they approximate random stochastic operators. Edelman–Sutton conjecture. Scaling li…

Essential spectrum conjecture. Almost surely,

Let $H_{\rm Anderson}=-\lambda\Delta+V$ be the Anderson model on the lattice $\mathbb{Z}^d$ for $d\ge 3$, where $\Delta$ is the discrete Laplacian, $V$ is a random potential with i…

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 a regular tree be given, let a finite graph be given, and consider the Anderson model obtained by adding a random potential to the Laplace operator on their product. The spectr…

Critical-exponent conjecture. At both of these critical couples, the limit of the normalized logarithmic growth is probably infinite for $\nu<1/2$ and zero for $\nu>1/2$. This conj…

For a random operator with disorder, consider the energy levels of finite-volume restrictions on the scale of their typical energy spacing. Write “pp” for pure-point spectral regim…

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…

For the random dimer model, let $\delta_1$ denote the initial state at site $1$, let $\beta_{\delta_1,\omega}^-(2)$ be its lower transport exponent of order $2$ for a realization…

Diffusion-pole conjecture. One has

Let $H$ be the Landau Hamiltonian with random potential $V$, where the magnetic field is uniform with strength $1$. A potential is regarded as sufficiently random in the sense inte…

Let the integrated density of states (IDS) be the spectral distribution measure associated with a Schrödinger operator having a random potential. Lifshitz's conjecture. The IDS is…

Consider the one-dimensional Anderson–Bernoulli random dimer model on $\ell^2(\mathbb{Z})$, with potential values $\pm V$ arranged in independent identically distributed dimers, wh…

Kleptsyn–Quintino conjecture. In the independent case, quasi-everywhere convergence in the displayed relation fails.

Strong-disorder MPDLE conjecture. MPDLE holds in the regime of strong disorder. This conjecture concerns the decay of multi-point correlation functions and would extend two-point d…

Let $H=H_\beta=-\beta\Delta+V$ be the Anderson Hamiltonian on a discrete torus $\Lambda$ approximating $\mathbb{Z}^d$, and let … In the same setting, the more precise quantum diffu…

Anderson delocalisation conjecture. For the Anderson model, this occurs in $d\geq3$ for $E$ inside the spectrum of $-\beta\Delta$ when $\beta$ is sufficiently large. This is a long…

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 a random operator have spectral regions in which eigenfunctions exhibit localization or delocalization. Local eigenvalue statistics describe the rescaled eigenvalue point proce…

The Anderson Hamiltonian $d539^{(E)}$ has eigenvectors whose rescaled and rotated profiles are described by the limiting critical stochastic operator $d4c2\mathcal{S}_\tau$. More p…

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 $H$ be the multi-dimensional Anderson Hamiltonian, with dimension $d\geq 3$. At sufficiently low energies, the extended-states conjecture. $H$ has extended states; equivalently…

Let $H_\alpha=(-\Delta)^\alpha+\lambda V_\omega$ be the fractional Anderson model in dimension $d=1$, with fractional parameter $\alpha$. Phase-transition conjecture. As $\alpha$ v…

Let $V_j$ be non-negative i.i.d. random variables with distribution $\mu$ such that $0 \in \operatorname{supp} \mu \not= \{0\}$. Let $K\in \mathbb{N}$ and let $\delta_0<1$. For nor…