Consider the usual lattice of $\mathbb{Z}^2$ where we simply add in the positive diagonals.
That is the graph $G = (V,E) $ where $V = \mathbb{Z}^2$ and $E$ consists of edges where we connect points $(x_1,y_1) $ and $(x_2,y_2)$ if either: $x_1 = x_2$ and $y_1 = y_2 +1$
or $x_1 = x_2 +1$ and $y_1 = y_2$
or $x_1 = x_2 +1$ and $y_1 = y_2 +1$
I wish to show that $p_C(G) \leq \frac{1}{2}$ without using the exponential decay theorem.
A little context around the problem and my attempts
The first part of this question nudged around the dual of the graph and in it I showed that each vertex of the dual of $G$ has degree 3. The question is apparently meant to be fairly simple and, if this is at all relevant, was only worth 5 marks, (the paper has 60 in total)
I have tried to bound the probability of an infinite cluster at the origin by an infinite sum of probabilities of minimal cuts surrounding the origin and then with a little graph theory got elements of this sum in terms of cycles surrounding the origin but have not made it much further that this.
Any help would be great!