The probability that n random d-dimensional lattice vectors are pairwise equidistant is asymptotically an explicit moment-dependent constant divided by d^{(n(n-1)/2 - 1)/2} as d grows.
Gross, Jay Yellen, and Mark Anderson
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On the probability of n equidistant points in high-dimensional lattices
The probability that n random d-dimensional lattice vectors are pairwise equidistant is asymptotically an explicit moment-dependent constant divided by d^{(n(n-1)/2 - 1)/2} as d grows.