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

2026-05-29

AI Summary

This work refutes Steurer’s conjecture that any family of unit vectors with low pairwise correlations must contain a constant-separated subset of linear size. By constructing a counterexample based on sparse high-dimensional expanders, the authors demonstrate that even when the average correlation among vectors is polynomially small, no linear-sized approximately separated subset may exist. The proof hinges on three key technical contributions: the design of high-dimensional expanders, a refined analysis of vector correlations, and an estimate of the $L_2$ mixing time for reweighted random walks. As a byproduct, this construction yields the first example of a vertex expander on which every reweighted simple random walk exhibits an $L_2$ mixing time of at least $\log^{5/4 - o(1)} n$, thereby revealing a fundamental limitation in the relationship between correlation and separability.

Abstract

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

Problem

• High-Dimensional Expanders
• Sparsest Cut Problem
• Steurer's Conjecture
• Vector Correlation
• Separated Sets

Innovation

• high-dimensional expanders
• sparsest cut
• Steurer's conjecture
• L2-mixing time
• random walks