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@MartinBrandenburg: Worse, now they're showing ads instead. I'm going to edit them out; hopefully the original author can replace them with properly hosted versions.
I believe some statements equivalent to this property (assuming that $||(x,y)||=n(x,y)$ is a norm on $\mathbb R^2$) are a) that $||(x,y)||_{\rm abs} = n(|x|,|y|)$ is a norm on $\mathbb R^2$, b) that $n(x,y) \ge n(x,0), n(0,y)$ for all $x,y\ge0$, or c) that the balls $B_{\rm abs}(r)=\{(x,y)\in\mathbb R^2:n(|x|,|y|)\le r\}$ are convex.
What you have is basically a contact process on a weighted graph, so you may find the theory of such processes useful (especially if you want to consider the limit $N\to\infty$). But, as lmg notes, for small $N$ you can just write down the explicit transition matrix for the combined process and exponentiate it.
