论文标题
板岩指标的随机近似
Stochastic approximation of lamplighter metrics
论文作者
论文摘要
我们观察到，可以将嵌入到随机指标中的嵌入可以有效地用于研究lamplighter图或组的$ l_1 $ - 内置性，以及更通常的灯塔指标空间。一旦建立了这种连接，对$ l_1 $ distortion的lamplighter指标的几项新的上限估计值是从有关随机嵌入到主导的树量表中的已知相关估计值。例如，$ n $ - 点公制空间上的每个Lamplighter指标都将Bi-lipschitzly嵌入$ L_1 $中，并带有变形$ O（\ log n）$。特别是，对于每个有限组$ g $，lamplighter组$ h = \ mathbb {z} _2 \ wr g $ bi-lipschitzly嵌入$ l_1 $，带有变形$ O（\ log \ log \ log | h |）$。如果灯塔结构中的地面空间是具有一些拓扑限制的图表，则可以实现更好的失真估计。最后，我们讨论如何将粗嵌入到$ d $ d $ d $二维的无限格子$ \ mathbb {z}^d $上构建的lamplighter group，可以从limplighter图的bi-lipschitz嵌入式中构建。
We observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the $L_1$-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on a $n$-point metric space embeds bi-Lipschitzly into $L_1$ with distortion $O(\log n)$. In particular, for every finite group $G$ the lamplighter group $H = \mathbb{Z}_2\wr G$ bi-Lipschitzly embeds into $L_1$ with distortion $O(\log\log|H|)$. In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into $L_1$ of the lamplighter group over the $d$-dimensional infinite lattice $\mathbb{Z}^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite $d$-dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.
