发表机构
Qiuzhen College, Tsinghua University（清华大学求真书院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究异步$\ell^p$松弛的共识时间问题，针对盒图、树、电导扩张器等不同图结构，推导得到依赖图性质的估计结果，明确了各类图下共识时间的相关界。
AI 中文摘要
我们研究由阿米尔、纳扎罗夫和佩雷斯提出的异步$\ell^p$松弛：每一步中，随机均匀选取的一个顶点会使其关联的$\ell^p$能量最小化。对于经过$t$次更新后的分布$f_t$，令$$ \mathsf T_p(G,1/2):= \sup_{\lVert f_0\rVert_\infty\le1} \mathbb{E}\bigl[\min\{t\ge0:\operatorname{osc}(f_t)\le1/2\}\bigr]. $$。针对$1<p<\infty$的情形，我们得到了盒图、树以及电导扩张器的依赖图性质的估计结果。在最近邻盒图$[L]^d$上，对于$d,L\ge2$和$n=L^d$，在忽略对数因子的情况下，结果为：当$1<p<2$时为$nd^{1/(p-1)}L^{p/(p-1)}$，当$p\ge2$时为$ndL^2$。在有界度树上，一个显式的重根不变参数$T_G$可在对数因子精度内确定结果；对于任意树，该参数给出的上下界还额外相差最大度因子。若体积电导为$h(G)\ge h_0>0$，则无需度的假设即可得到$\mathsf T_p(G,1/2)=Θ_{p,h_0}(n\log n)$。当处于$p=\infty$时，每个连通图都满足$\mathsf T_\infty(G,1/2)\ge c nD^2/Δ$，其中$D$和$Δ$分别为图的直径和最大度。
英文摘要
We study the asynchronous $\ell^p$ relaxation introduced by Amir, Nazarov, and Peres: at each step, a uniformly chosen vertex minimizes its incident $\ell^p$ energy. For the profile $f_t$ after $t$ updates, let $$ \mathsf T_p(G,1/2):= \sup_{\lVert f_0\rVert_\infty\le1} \mathbb{E}\bigl[\min\{t\ge0:\operatorname{osc}(f_t)\le1/2\}\bigr]. $$ For $1<p<\infty$, we obtain graph-dependent estimates for boxes, trees, and conductance expanders. On the nearest-neighbor box $[L]^d$, for $d,L\ge2$ and $n=L^d$, the answer is, up to logarithmic factors, $nd^{1/(p-1)}L^{p/(p-1)}$ for $1<p<2$ and $ndL^2$ for $p\ge2$. On bounded-degree trees, an explicit rerooting-invariant parameter $T_G$ determines the answer up to logarithmic factors; for arbitrary trees it gives upper and lower bounds that differ additionally by the maximum degree. If the volume conductance $h(G)\ge h_0>0$, then $\mathsf T_p(G,1/2)=Θ_{p,h_0}(n\log n)$ without a degree assumption. At $p=\infty$, every connected graph satisfies $\mathsf T_\infty(G,1/2)\ge c nD^2/Δ$, where $D$ and $Δ$ are its diameter and maximum degree.
Comments35 pages
