53 problems

Super-Brownian scaling conjecture. The critical-dimensional model should have the same super-Brownian scaling limit as the high-dimensional model once appropriate logarithmic corre…

Pointwise upper-bound conjecture. The pointwise upper bound

Transience conjecture. Supercritical long-range percolation is transient when $d\geq 3$ and $\alpha\geq 2d$.

Largest-cluster size conjecture. As $r\to\infty$,

Let $d<6$ and $b1=d/3$. Consider long-range percolation on $Z^d$ with parameter $b1$. Logarithmic-correction conjecture. The same logarithmic corrections computed for hierarchical…

Constant mixing-time conjecture. The mixing time $\tau(G_{s,\beta}(N))$ is constant, independent of $N$.

Long-range percolation scaling conjecture. If $\alpha\in(0,2)$, then the corresponding random walk scales to a symmetric $\alpha$-stable Lévy process in $\mathbb R^d$.

Supercritical upper-bound conjecture. The displayed limsup is finite.

Finiteness conjecture. The corresponding directional limit should be finite (on the connected subsequence when nearest-neighbor bonds are not forced to exist).

Let $G$ be the random graph on vertex set ${\mathbb Z}$ in which each pair of vertices $u,v$ is joined independently with probability $p_{|u-v|}$, where $p_1=1$ and…

Consider the long-range percolation graph on the vertex set $\{0,1,\ldots,N\}^d$, where vertices at distance $r$ are connected with probability approximately $\beta/r^s$. Linear di…

Consider the one-dimensional long-range percolation graph on the finite circle with vertex set $\{0,1,\ldots,N\}$, in which neighboring vertices are joined with probability one and…

Consider the one-dimensional long-range percolation graph on the finite circle with vertex set $\{0,1,\ldots,N\}$. Neighboring vertices are joined with probability one, and distinc…

Critical-exponent conjecture. (A) If $s=2$, then the diameter's order of magnitude is $N^\delta$, where $\delta$ is a function of $\beta$. (B) If $1<s<2$, then the diameter is…

Critical-kernel conjecture. The hierarchical-lattice trichotomy has an analogous form on $\mathbb{Z}$ when $\|x-y\|^{-2d}\log\log\|x-y\|$ is replaced by $|x-y|^{-2}$; in particular…

Imbrie–Newman conjecture. The lower bound for the density of the infinite cluster is an equality:

Long-range percolation crossover conjecture. Such a crossover value exists, with the regime $\alpha>\alpha_c$ belonging to the same universality class as the nearest-neighbour mode…

Non-concavity conjecture. There exists $\delta=\delta(k,d)>0$ such that $p\mapsto q_c(p)$ is not concave on $[0,\delta]$.

Hutchcroft's Erdős–Rényi universality conjecture. This model is in the Erdős–Rényi universality class when

Derivative-comparison conjecture. For $r\geq1$ and $\beta_c/2\leq\beta\leq\beta_c$,

Essam–Gaunt–Guttmann logarithmic-correction conjecture. The same logarithmic corrections to scaling should appear in the long-range model with $d=6$ and $\alpha>2$.

Let $d$ be the dimension, let $\beta$ be the percolation parameter, and let $r$ denote the radius variable in the model. Write $\rho_p(\beta,r)$ for the relevant radius of gyration…

Pointwise multipoint-asymptotics conjecture. These diagrammatic estimates should hold pointwise, in the sense of first-order asymptotics, whenever all distances between the points…

Continuity conjecture. The phase transition is continuous for every $\alpha>0$ when $d\geq 2$, meaning that no infinite cluster exists at $\beta=\beta_c$. The transition is known t…

Positivity conjecture. If $s>2$, then