Locally finite connected graphs admit coarse embeddings into Hilbert space if and only if they support bond percolations with arbitrarily large marginals and two-point function vanishing at infinity, with an analogous characterization of the L1-compression exponent via stretched-exponential decay.
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Coarse embeddability, $L^1$-compression and Percolations on General Graphs
Locally finite connected graphs admit coarse embeddings into Hilbert space if and only if they support bond percolations with arbitrarily large marginals and two-point function vanishing at infinity, with an analogous characterization of the L1-compression exponent via stretched-exponential decay.