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arXiv · 2606.00292

High-Dimensional Expanders, the Sparsest Cut Problem, and Steurer's Conjecture

Farzam Ebrahimnejad·Shayan Oveis Gharan
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Abstract

In 2010, Steurer conjectured that any family of $n$ unit-norm vectors $v_1,\dots,v_n$ with polynomially small average correlation $\mathbb{E}_{i,j}|\langle v_i,v_j\rangle|\leq n^{-\epsilon}$ contains linear-sized constant-separated sets. We refute this conjecture in a strong sense using the machinery of sparse high-dimensional expanders: such vector families do not even have linear-sized $\frac{1}{\log^{1/4-o(1)}(n)}$-separated sets. Consequently, we show that there are families of vertex expanders on $n$ vertices for which the (average) $L_2$-mixing time to the uniform distribution of any reweighted simple random walk is at least $\log^{5/4-o(1)} n$.

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Farzam Ebrahimnejad, Shayan Oveis Gharan. 2026-05-29. High-Dimensional Expanders, the Sparsest Cut Problem, and Steurer's Conjecture. https://arxiv.org/abs/2606.00292

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