42 problems

Let an $N\times N$ random band matrix (RBM) have bandwidth $W$. The eigenvectors may be completely delocalized for large bandwidth and exponentially localized for small bandwidth,…

Let $A$ be a Gaussian random band matrix with bandwidth $W$, normalized so that its eigenvalues $E$ are concentrated in $[-2,2]$ and its asymptotic density of states is expected to…

Let $R_k$ be the $k$-point correlation function of the eigenvalues of an $N\times N$ Hermitian random matrix, let $E$ lie in the bulk of the spectrum, let $\rho(E)$ denote the limi…

Localization-delocalization threshold conjecture. The threshold for the localization-delocalization transition is conjectured to satisfy

Let $\Lambda_L$ be the index set of the random band matrix, let $\sigma_{xy}$ denote its variance-profile entries, and let $H_{xy}$ denote the matrix entries. The coefficient $a_4$…

Let $W$ denote the band width and let $\ell$ denote the localization length of a two-dimensional random band matrix; let $L$ be the linear size of the system and $W_c$ its critical…

Higher-dimensional localization length conjecture. The localization length is of order $N$:

Higher-dimensional PRBM regime conjecture. This model exhibits three distinct regimes according as $\alpha<0$, $0<\alpha<d$, or $\alpha>d$.

PRBM localization length conjecture. The localization length satisfies

Wave-packet propagation scale conjecture. The characteristic spatial scale of wave-packet propagation is

Exponential singularity estimate conjecture. An exponential-type estimate on the singularity probability of $A_n$ can be proven whenever

Let $W$ denote the bandwidth of an $N\times N$ random band matrix, interpolating between a one-dimensional random Schrödinger operator when $W\sim 1$ and a Wigner matrix when…

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…

Consider random band matrices with non-periodic variance profiles, including i.i.d. entries and general profile functions, in the regime $1\ll W\ll L$. Let Theorems … denote the ma…

Consider a random band matrix of size $N$ and bandwidth $W$. In the regime $W\sim O(1)$ with $N\to\infty$, the model approaches a finite-volume one-dimensional Anderson model with…

Scalar-fluctuation upper-bound conjecture. If $\varepsilon>0$ is small, $|\lambda|<\varepsilon$, $M\geq C_\varepsilon$, and $1<k<N$, then

Non-Gaussian extension conjecture. The use of the Gaussian law can be removed, and the same matrices treated by Schenker can also be handled using the methods of this paper.

Localization–delocalization transition conjecture. For small $\varepsilon>0$ and sufficiently large $N\geq C_\varepsilon$, the eigenvectors of $A$ are typically localized when…

Random band matrices are matrices whose nonzero entries are restricted to a band of a given width. The Poisson-to-random-matrix transition conjecture asserts that there is a transi…

Localization-delocalization transition conjecture. There exists $W_c\in[1,L]$ such that if $W\ll W_c$, the bulk eigenvectors of $H$ are localized, whereas if $W\gg W_c$, they are d…

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 $W$ denote the band width and let $\ell$ denote the localization length of a one-dimensional random band matrix; let $L$ be the linear size of the system and $W_c$ its critical…

Let $H$ be an $N\times N$ random band matrix with bandwidth $W$. The eigenvectors of $H$ and the local spectral statistics are expected to depend on the bandwidth: for large $W$, t…

Let a one-dimensional random band matrix (RBM) have size $N$ and bandwidth growing like $N^\alpha$, with $0\leq\alpha\leq 1$. Critical bandwidth conjecture. The critical bandwidth…

Consider random Hermitian band matrices of size $N$ and bandwidth $W$. Their local bulk eigenvalue statistics are expected to transition from Poisson statistics to GOE/GUE statisti…