17 problems

Let … where the $U_j(x)$ are fixed non-random potentials of class $C^1$ satisfying condition, and $(a_j^\omega(t),1\leq j\leq N)$ is a realization of a stationary vector-valued ran…

In the perfect Bose gas with repulsive impurities, type I Bose–Einstein condensation means that the condensate is macroscopically occupied only in the ground state. Kac and Lutting…

The model involves a Poisson point process with random potential values whose order statistics determine the variational functional governing the walk. The functional $\Xi(\Pi)$ is…

Let $d>1$, let $u_{\text{\tiny\rm gs}}$ be the ground state of the Gross–Pitaevskii eigenvalue problem (GPEVP), and let $V_\varepsilon$ be a disorder potential satisfying assumptio…

Let $d>1$, and let $u_{\text{\tiny\rm gs}}$ denote the ground state of the Gross–Pitaevskii eigenvalue problem (GPEVP). Higher-dimensional ground-state bound. The ground state sati…

Almost Mathieu diffusion conjecture. For almost every $\theta,\alpha\in[0,1]$, the equation has a diffusion constant $D(g,\lambda)\in(0,\infty)$ that is smooth for all…

Diffusive propagation and disorder-scaling conjecture. For every bounded potential $u$ and every $\lambda>0$, these constants are positive and finite, with…

Let $r$ denote the depinning rate, and let $s\ll 1$ be the parameter corresponding to the large-mass limit. The perturbative generalized Lyapunov exponent is expanded in powers of…

Let $cxi(z)=cxi(-z)$ with probability $p=p(|z|)$, where $p(n)cto1$ as $nctocinfty$, and fix a Pareto parameter $calpha>2$. Complete localisation is the property that the parabolic…

Almost-sure diffusion conjecture. If $g>0$ and $\psi_0$ is normalized in $\ell^2(\mathbb Z^d)$, then with probability one $\psi_t(x)$ evolves diffusively:…

Let $d\geq 4$ and let $\mu_{L,E}^0$ be the measures associated with the finite-volume model at energy $E\in(-2d,2d)$. Let $n_d$ denote the density of states of the $d$-dimensional…

Upper-energy conjecture. When the integrability condition on $V$ is dropped, the upper energy $\beta\mathbb{E}[V]$ in the corresponding localization result should be replaced by

In the directed i.i.d. case in dimension $d=2$, consider a point-to-point free energy in the KPZ universality class, centered and normalized according to its fluctuation scale. KPZ…

Consider a Schrödinger operator $H=-\Delta+V$ on a unit square, where $V$ is a random potential, and let $u$ solve … with Dirichlet boundary conditions. The valley network is obtai…

Let a homogeneous random potential contain an interacting Bose gas, and consider generalized Bose–Einstein condensation in this random medium. The Kac–Luttinger conjecture. Condens…

Consider a system with a temperature parameter $T$ and a duality transformation $T\mapsto 1/T$. A function is self-dual if it is invariant under this transformation. Freezing-trans…

Let $\xi$ be the random potential satisfying the essential-supremum and bounded independent-identically-distributed assumptions, let $\kappa>0$ be the diffusion parameter, and let…