14 problems

Outside-cone fluctuation conjecture. If

Fix two distinct source sites in $\mathbb{Z}^d$, and suppose $0\le p\le q\le1$. For $i=1,2$, let $\mathcal{G}^i_{p,q}$ be the event that infection $i$ survives indefinitely: … Let…

Let $p_c(d,R)$ denote the critical probability for the existence of an infinite connected cluster in the directed space-time percolation model with spatial range $R$, in dimension…

Let $\Gamma_s^\epsilon$ be the set of right-most infinite open paths at dynamical time $s$. Let $S_{\alpha(p),\sigma(p),\epsilon}$ denote the corresponding space-time rescaling, an…

Barrier-cone characterization conjecture. For every $p\in[0,1]$,

Consider sufficiently spread-out oriented percolation in dimension $d>4$, with spread parameter $L$ sufficiently large, and let $\mathcal {T}_n$ denote the set of vertices connecte…

Let the authors' general conditions be the hypotheses under which rescaled ranges of a single occupancy particle model converge to the range of super-Brownian motion. Range-scaling…

For $0 \leq p \leq 1$, let $\vec{\mathbb{Z}}^2(p)$ be the random orientation of the square lattice in which each horizontal edge is oriented rightwards with probability $p$ and lef…

Kordzakhia's limiting-shape conjecture. The influence region has a definite limiting shape $S\subset {{\mathbb R}}^2$ in the sense that, almost surely, $R_t$ converges to the shape…

Fix $p\text{ in }(p_c,1)$, let $\Gamma$ be the collection of all rightmost infinite open paths in the supercritical oriented percolation configuration, and let $\alpha=\alpha(p)>0$…

Asymptotic shape conjecture. For every $p>\overrightarrow{p_c}^{\text{alt}}(d+1)$, every $q<\overrightarrow{p_c}^{\text{alt}}(d+1)$, and every $\alpha\in(0,\alpha_c(p,q))$, there e…

Oriented-percolation critical crossing summability conjecture.

Oriented-percolation anisotropic crossing conjecture. There exist $z$, $c^u_{OP}$, $c^l_{OP}$, and $\xi(\rho)$ such that $0<z<1$, $0<c^u_{OP}<1$, $0<c^l_{OP}<1$, $\xi(\rho)<\infty$…

Knight-model transition conjecture. The transition for the Knight model occurs at $p_c^{OP}$.