Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stochastic approximation of lamplighter metrics   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.

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The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn