Phase transition for percolation on a randomly stretched lattice

TL;DR

This paper studies a percolation model on a randomly stretched lattice, showing that the existence of a phase transition depends on the finiteness of certain moments of the stretching variables.

Contribution

It establishes a precise relationship between phase transition occurrence and the moment properties of the random edge lengths in the model.

Findings

Phase transition occurs if (\xi_1^) <  for some  > 1.

No phase transition if (\xi_1^) =  for some  < 1.

The phase transition behavior is characterized by the moments of the stretching distribution.

Abstract

Let $\{\xi_i\}_{i \geq 1}$ 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 $\xi_i$. Then declare independently each edge $e$ in the resulting lattice open with probability $p_e=p^{|e|}$ where $p\in[0,1]$ and $|e|$ is the length of $e$. We relate the occurrence of nontrivial phase transition for this model to moment properties of $\xi_1$. More precisely, we prove that the model undergoes a nontrivial phase transition when $\mathbb{E}(\xi_1^\eta)<\infty$, for some $\eta>1$ whereas, when $\mathbb{E}(\xi_1^\eta)=\infty$ for some $\eta<1$, no phase transition occurs.