Coarse embeddability, $L^1$-compression and Percolations on General Graphs

TL;DR

This paper characterizes when a locally finite, connected graph can be coarsely embedded into a Hilbert space using probabilistic percolation methods, linking decay of the two-point function to $L^1$-compression.

Contribution

It introduces new probabilistic techniques to characterize coarse embeddability and $L^1$-compression exponents for general graphs, extending previous group-invariant methods.

Findings

Coarse embeddability is equivalent to existence of certain percolations with large marginals.

Decay of the two-point function relates to the $L^1$-compression exponent.

Results apply to Cayley graphs of finitely generated groups and beyond.

Abstract

We show that a locally finite, connected graph has a coarse embedding into a Hilbert space if and only if there exist bond percolations with arbitrarily large marginals and two-point function vanishing at infinity. We further show that the decay of the two-point function is stretched exponential with stretching exponent $\alpha\in[0,1]$ if and only if the $L^1$-compression exponent of the graph is at least $\alpha$, leading to a probabilistic characterization of this exponent. These results are new even in the particular setting of Cayley graphs of finitely generated groups. The proofs build on new probabilistic methods introduced recently by the authors to study group-invariant percolation on Cayley graphs [28,29], which are now extended to the general, non-symmetric situation of graphs to study their coarse embeddability and $L^1$-compression exponents.