Local and global expansion in random geometric graphs

@article{Liu2022LocalAG,
  title={Local and global expansion in random geometric graphs},
  author={Siqi Liu and Sidhanth Mohanty and Tselil Schramm and Elizabeth Yang},
  journal={ArXiv},
  year={2022},
  volume={abs/2210.00158}
}
Consider a random geometric 2-dimensional simplicial complex X sampled as follows: first, sample n vectors u1, . . . ,un uniformly at random on S d−1; then, for each triple i, j,k ∈ [n], add {i, j,k} and all of its subsets to X if and only if 〈 ui ,u j 〉 Ê τ,〈ui ,uk〉 Ê τ, and 〈 u j ,uk 〉 Ê τ. We prove that for every ε > 0, there exists a choice of d = Θ(logn) and τ = τ(ε,d) so that with high probability, X is a high-dimensional expander of average degree n in which each 1-link has spectral gap… 

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