Random 0/1 polytopes have edge-expansion Θ(d) whp for p ≤ 1-ε and Ω(d^k) for any k when p ≤ 1/2-ε, verifying the Mihail-Vazirani conjecture in strong form with a phase transition at p=1/2.
arXiv preprint arXiv:2311.07210 , year=
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Proves that the giant component in hypercube bond percolation at p = c/d > 1 has diameter Θ(d) and lazy random walk mixing time Θ(d²).
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The Mihail-Vazirani conjecture and strong edge-expansion in random $0/1$ polytopes
Random 0/1 polytopes have edge-expansion Θ(d) whp for p ≤ 1-ε and Ω(d^k) for any k when p ≤ 1/2-ε, verifying the Mihail-Vazirani conjecture in strong form with a phase transition at p=1/2.
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Diameter and mixing time of the giant component in the percolated hypercube
Proves that the giant component in hypercube bond percolation at p = c/d > 1 has diameter Θ(d) and lazy random walk mixing time Θ(d²).