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Let $G$ be Green's function for the simple random walk on $\mathbb Z^n$ for $n\ge 3$, i.e., $G(x)$ is the expected number of visits to $x$ when the walk starts at the origin.

Define $d(x,y)=G(x-y)^{1/(2-n)}$ for $x\ne y$ and $d(x,x)=0$. This is asymptotic to a multiple of the Euclidean metric for $x$ and $y$ far apart (since $G(x) = \beta_n |x|^{2-n} + O(|x|^{-n})$). But is $d$ a metric? (The case $n=3$ is of special interest.)

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  • $\begingroup$ How far have you numerically computed values of $d(0,x)$ for $n=3$? I guess that the likeliest candidates for failure of the triangle inequality are degenerate or nearly degenerate lattice triangles. $\endgroup$ Commented Oct 11, 2023 at 18:27
  • $\begingroup$ Why do you need a metric defined in terms of the Green function ? I'm asking because I don't think that your function $d$ is a true metric, but maybe you are not aware that $-\log G$ is a metric and this would help you. $\endgroup$ Commented Apr 18, 2024 at 11:13
  • $\begingroup$ It's a long story having to do with some philosophical stuff about theories of discrete space, and yes $-\log G$ doesn't do the job. I wanted a metric naturally definable in terms of random walks and asymptotic to the Euclidean metric. $\endgroup$ Commented Apr 19, 2024 at 3:26

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