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

Abstract

In 2010, Steurer conjectured that any family of $n$ unit-norm vectors $v_1,\dots ,v_n$ with polynomially small average correlation ${\mathbb{E}}_{i,j}|⟨v_i,v_j⟩|\le n^{-\hat{I}\mu }$ 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 ${\log }^{5/4-o(1)}n$.

10 pages