Crossing Bounds and Shape Functions in Anisotropic Percolation

Definitions and mathematical statement

The phrase "the same model" is clarified by the adjacent local record OWR-1731-004: bond percolation on the square lattice ${\mathbb{Z}}^2$, with horizontal and vertical open-edge probabilities $p_h,p_v$ satisfying $p_h+p_v=1$. In the conventional independent-edge interpretation, write $\Pr_{p_h,p_v}$ for the product law and let $H(a,b)$ be the event that an open nearest-neighbor path in the lattice rectangle $[0,a]\times[0,b]$ joins its left and right sides.

The question asks whether the known rectangular a priori crossing bounds can be improved to give stronger asymptotics for the report's aspect-ratio or shape function $f(n)$, in particular a nontrivial limit \[ \lim_{n\to\infty}\frac{n}{f(n)}. \] A positive finite limit would be one natural reading of "nontrivial," but the extraction does not define that word or specify a proposed value. More broadly, it asks for a class of shapes admitting both an a priori crossing bound and a conditional extension theorem.

Neither this record nor the adjacent model record defines $f(n)$, states the existing rectangular estimate, or specifies the conditioning event and conclusion of the extension theorem. Thus no particular threshold definition, rectangle dimension convention for $f(n)$, rate, lower-bound constant, or conditional probability inequality is inserted here. It is also unclear whether parameters are fixed and constants may depend on them, or whether uniformity over anisotropies is intended; the nondegenerate range $0<p_h,p_v<1$ is not explicitly stated. These data are required for a checkable asymptotic or crossing theorem.

Short English statement

Can crossing estimates in critical anisotropic square-lattice percolation yield a nontrivial limit of $n/f(n)$, and can suitable shapes support both an a priori bound and a conditional extension result? The function and estimates are not defined in the extraction.

Sources

• Local UnsolvedMath registry, permanent ID $30000860$, source code catalogue entry, "Crossing Bounds and Shape Functions in Anisotropic Percolation." No individual proposer is supplied.
• Adjacent local registry record, permanent ID $30000859$, OWR-1731-004, "RSW Theory for Anisotropic Square-Lattice Percolation," supplies the probabilities $p_h+p_v=1$ and the event $H(a,b)$ referred to by "the same model." It does not supply a definition of $f(n)$.
• Arbeitsgemeinschaft: Percolation (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/48, pp. 2851--2892: https://doi.org/10.4171/owr/2007/48. This is the sole supplied external reference.
• Supplied literature triage, checked 2026-08-21: status "open." It describes this as a deliberately broad companion shape-function question and reports no identified general resolution. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.