High-Dimensional Expanders, the Sparsest Cut Problem, and Steurer's Conjecture
This resolves a conjecture in high-dimensional expander theory, providing a strong counterexample with implications for spectral graph theory and random walks.
The authors refute Steurer's conjecture on vector families with small average correlation, showing that such families cannot have linear-sized constant-separated sets, and imply that certain vertex expanders have slow mixing times of at least log^{5/4-o(1)} n.
In 2010, Steurer conjectured that any family of $n$ unit-norm vectors $v_1,\dots,v_n$ with polynomially small average correlation $\mathbb{E}_{i,j}|\langle v_i,v_j\rangle|\leq n^{-ε}$ 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 $\log^{5/4-o(1)} n$.