A Strict Inequality for the Random Triangle Model

摘要

The random triangle model on a graph G, is a random graph model where the usual i.i.d. measure is perturbed by a factor q t(ω), where q≥1 is a constant, and t(ω) is the number of triangles in the random subgraph ω. Here we consider the case where G is the usual two-dimensional triangular lattice, for which there exists a percolation threshold p c (q) such that the probability of getting an infinite connected component of retained edges is 0 for p c (q), and 1 for p>p c (q). It has previously been shown that p c (q) is a decreasing function of q. Here we strengthen this by showing that p c (q) is strictly decreasing. This confirms a conjecture by Häggström and Jonasson.