AI 中文总结
本文通过构造叶子连接到完全图的图，反驳了重复平均过程 $L^2\\!\to\\!L^1$ 混合时间阶为 $|E|\log n/\lambda_2$ 的猜想，证明其混合时间为 $\Theta(n^2)$ 且无对数因子，并揭示特征向量强局部化现象。
AI 中文摘要
重复平均过程是图上的一个随机平均过程，其混合时间在若干结构化图族中已知，但尚未对所有连通图给出一般性的精确表达式。曾有猜想认为 $L^2\\!\to\\!L^1$ 混合时间阶为 $|E|\log n/\lambda_2$。我们利用将一片叶子连接到完全图 $K_n$ 上得到的图 $G_n$ 反驳了这一猜想。我们证明 $\lambda_2(G_n)=1$ 且 \\[ t_{\varepsilon,2\to1}(G_n)=\Theta_{\varepsilon}(\gamma(G_n))=\Theta_{\varepsilon}(n^2), \\] 没有额外的 $\log n$ 因子。该例子具有强局部化的 Fiedler 特征向量：其平方 $L^2$ 质量大部分位于叶子上，而平衡质量则分布在团上。
英文摘要
The repeated averages process is a stochastic averaging process on graphs whose mixing time is known for several structured families, but no general sharp expression is known for all connected graphs. It was conjectured that the $L^2\!\to\!L^1$ mixing time is of order $|E|\log n/λ_2$. We disprove this conjecture using the graph $G_n$ obtained by attaching one leaf to the complete graph $K_n$. We prove that $λ_2(G_n)=1$ and that \[ t_{\varepsilon,2\to1}(G_n)=Θ_{\varepsilon}(γ(G_n))=Θ_{\varepsilon}(n^2), \] with no additional $\log n$ factor. The example has a strongly localized Fiedler eigenvector: most of its squared $L^2$ mass lies on the leaf, while the balancing mass is spread across the clique.
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