Factorization Formulas for $2D$ Critical Percolation, Revisited

Abstract

We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1,u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $\mathbb{P}(n{u}_1\leftrightarrow n{u}_2\leftrightarrow nw{)}^2/\mathbb{P}(n{u}_1\leftrightarrow n{u}_2)\cdot \mathbb{P}(n{u}_1\leftrightarrow nw)\cdot \mathbb{P}(n{u}_2\leftrightarrow nw)$ converges to $K_F$ as $n\to \infty$, where $x\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $\mathbb{P}(n{u}_2\leftrightarrow [n{u}_1,n{u}_1+s];nw\leftrightarrow [n{u}_1,n{u}_1+s])$, where $s>0$.

Final version. To appear in Stochastic Processes and their Applications

References in corpus (1)

• Exact factorization of correlation functions in 2-D critical percolation