12 problems

Let $p_L^{(c)}$ be the stationary coefficients in model C, and let $M_L^{(c)}$ be their maximal value. Conjecture 13. The maximal coefficient $M_L^{(c)}$ occurs in the set…

Let $u$ and $v$ be Ballot paths of lengths $L$ and $L+1$, respectively. Let $v^*$ be the Anchored Cross path of length $L$ obtained by right reduction of $v$, with heights defined…

Let $\{h_i\}$ be any configuration in model A, and let $p^{(a)}_L$ denote its stationary-state coefficient. Conjecture 6. For every configuration $\{h_i\}$, there exists a configur…

Let $p_L^{(a)}$ and $p_L^{(b)}$ be the stationary coefficients in models A and B, and let $M_L^{(a)}$ and $M_L^{(b)}$ be their maximal coefficients. Let $S_L^{(a)}$ be the model-A…

Let $U_0$ be the nonlinear interaction coefficient, let $\mu$ be the chemical potential, and let $(\mathcal{E}_n,\phi_n)$ be an eigenvalue and eigenfunction of the associated linea…

Numerical-error conjecture. The numerical stationary state appearing to exceed $\rho_E$ for values such as $m=2.1$ and $m=2.5$ is a numerical error, as the stationary state moves a…

Let $m>0$, and let $\mathcal{S}_m$ denote the set of stationary solutions with total mass $m$. Let $Q_{gs}$ be a ground state, and let $E[Q]$ denote the total energy of a stationar…

Consider the nonlinear Schrödinger evolution with perturbation $\eta|\psi_t|^2\psi_t$, where $\eta\in\mathbb R$ is the nonlinearity strength, and let $\tilde\eta$ be the negative t…

Even-sector stationary-state conjecture. For each $\hat\rho_e\ge 1$, $\hat\mu_e^{(\hat\rho_e)}$ is the unique stationary state of the SSM on $\hat X^*_e$ with density $\hat\rho_e$.

Even-sector uniqueness conjecture. For each $\hat\rho_e\ge 1$, the state $\hat\mu_e^{(\hat\rho_e)}$ is unique among ETIS states on $\hat X^*_e$ with density $\hat\rho_e$.

High-density ETIS conjecture. At high density, every ETIS state belongs to one of the families described for the F-SSEP; equivalently, there are no other ETIS states.

Non-translation-invariant stationary-state conjecture. The F-ASEP has non-translation-invariant stationary states whenever $p\ne 1/2$, that is, whenever the process is genuinely as…