Factorization Formulas for 2D Critical Percolation, Revisited

2015/02/15 by Rene Conijn, Conijn, Rene
Mathematics · #60K35 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60K35

paper · pdf · doi:10.48550/arxiv.1502.04387

Final version. To appear in Stochastic Processes and their Applications

arxiv created 2015/05/28 · arxiv updated 2015/05/29

Abstract

We consider critical site percolation on the triangular lattice in the upper half-plane. Let u₁, u₂ 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 ℙ(nu₁ ↔ nu₂ ↔ nw)² / ℙ(nu₁ ↔ nu₂)⋅ℙ(nu₁ ↔ nw)⋅ℙ(nu₂ ↔ nw) converges to K_F as n → ∞, where x↔ 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 ℙ(nu₂ ↔ [nu₁,nu₁+s]; nw ↔ [nu₁,nu₁+s]), where s>0.

Related

• Is critical 2D percolation universal?
• A proof of factorization formula for critical percolation
• Critical percolation: the expected number of clusters in a rectangle
• Near-critical percolation in two dimensions
• Limit of the Wulff Crystal when approaching criticality for site percolation on the triangular lattice