we-solve (beta)

Math-2336 · mathematics · Probability

RSW Theory for Anisotropic Square-Lattice Percolation

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
In critical anisotropic bond percolation on the square lattice, let horizontal and vertical edge probabilities $p_h,p_v$ satisfy $p_h+p_v=1$, and let $H(a,b)$ be the event of an open horizontal crossing of $[0,a]\times[0,b]$. Does an RSW theorem hold for general $p_h,p_v$, namely a uniform positive a priori bound for square crossings and a conditional extension bound for $H(kn,n)$ for every $k$? More generally, is there a class of shapes for which both properties hold?

Formulation references: [S1], [S2].

Short English statement

In critical anisotropic bond percolation on the square lattice, let horizontal and vertical edge probabilities $p_h,p_v$ satisfy $p_h+p_v=1$, and let $H(a,b)$ be the event of an open horizontal crossing of $[0,a]\times[0,b]$. Does an RSW theorem hold for general $p_h,p_v$, namely a uniform positive a priori bound for square crossings and a conditional extension bound for $H(kn,n)$ for every $k$? More generally, is there a class of shapes for which both properties hold?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2336-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2336

At q=1 the source covers isotropic critical square-lattice bond percolation (p_h=p_v=1/2), yielding crossing control in this restricted subcase. It does not cover general anisotropic p_h+p_v=1 or determine the requested general shape class. This entry records partial support; complete coverage of the question is not established by this catalogue.

RSW Theory for Anisotropic Square-Lattice Percolation

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. The later interface theorem extends the square-lattice FK claim through $q=4$, with the same SLE parameter relation and a separate complete nested-loop limit.

Mathematical statement of the claimed result

Result 1.
Fix $q\in[1,4]$, $p_q=\sqrt q/(1+\sqrt q)$, and \[ \kappa(q)=\frac{4\pi}{\arccos(-\sqrt q/2)}. \] Use the square-lattice random-cluster law with edge-subset weights $p_q^{|A|}(1-p_q)^{|E|-|A|}q^{k_\xi(A)}$, where $\xi$ wires one boundary arc and leaves the other free. For a bounded Jordan domain with two marked boundary points, approximate its boundary uniformly by simple lattice polygons, preserving the marks. The claim gives \[ \eta_n\ \Longrightarrow\ \operatorname{SLE}_{\kappa(q)} \] for the Dobrushin interface in uniform curve distance modulo increasing reparametrization. Under the free infinite-volume law, its full nested loop collection satisfies \[ \Gamma_\delta\ \Longrightarrow\ \text{nested whole-plane }\operatorname{CLE}_{\kappa(q)} \] in the spherical matching topology: all loops above a shrinking diameter threshold are matched with shrinking uniform spherical error. Both limits hold along every mesh sequence, with $q$ fixed.

Outline of the argument

The tiled comparison and angular-transfer method is supplemented by endpoint estimates at $q=4$. Boundary current tests identify the driving process, while full exploration and localization retain macroscopic loops; at $q=1$ this also yields the bond-percolation crossing formula.

Formal verification

Family 223 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

Go to contribution →