Phase transition for percolation on a randomly stretched lattice

2019/12/06 by Hilario, Marcelo R., Sá, Marcos, Sanchis, Remy +1 · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1912.03320

Abstract

Let \ξ_i\_{i ≥ 1} be a sequence of i.i.d. positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in i-th vertical column to another in the (i+1)-th vertical column by an edge having length ξ_i. Then declare independently each edge e in the resulting lattice open with probability p_e=p^|e| where p∈[0,1] and |e| is the length of e. We relate the occurrence of nontrivial phase transition for this model to moment properties of ξ₁. More precisely, we prove that the model undergoes a nontrivial phase transition when 𝔼(ξ₁^η)1 whereas, when 𝔼(ξ₁^η)=∞ for some η<1, no phase transition occurs.

Cited by

• Phase transition on randomly horizontally stretched square lattice

Related

• A new proof for percolation phase transition on stretched lattices
• Phase transition on randomly horizontally stretched square lattice
• Constrained-degree percolation in random environment
• Sharp phase transition for percolation with short-range dependencies
• Merging percolation and classical random graphs: Phase transition in dimension 1