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 $\log^{5/4-o(1)} n$.

翻译：2010年，Steurer猜想：任意一组单位范数向量$v_1,\dots,v_n$，若其平均相关性$\mathbb{E}_{i,j}|\langle v_i,v_j\rangle|\leq n^{-ε}$呈多项式级小量，则该组向量必包含线性规模的恒定分隔集。我们利用稀疏高维展开子机制强否定这一猜想：此类向量族甚至不存在线性规模的$\frac{1}{\log^{1/4-o(1)}(n)}$分隔集。由此证明：存在$n$个顶点的顶点展开子族，使得任意重加权简单随机游走收敛至均匀分布的平均$L_2$混合时间至少为$\log^{5/4-o(1)} n$。