A strict inequality for the random triangle model

The random triangle model on a graph G, is a random graph model where the u sual i.i.d. measure is perturbed by a factor q(t(omega)), where q greater t han or equal to 1 is a constant, and t(omega) is the number of triangles in the random subgraph omega. Here we consider the case where G is the usual two-dimensional triangular lattice, for which there exists a percolation th reshold p(c)(q) such that the probability of getting an infinite connected component of retained edges is 0 for p < 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 co nfirms a conjecture by Haggstrom and Jonasson.