[2406.04222] Coarse embeddability, $L^1$-compression and Percolations on General Graphs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2406.04222 (math) [Submitted on 6 Jun 2024 ( v1 ), last revised 29 Apr 2026 (this version, v3)] Title: Coarse embeddability, $L^1$-compression and Percolations on General Graphs Authors: Chiranjib Mukherjee , Konstantin Recke View a PDF of the paper titled Coarse embeddability, $L^1$-compression and Percolations on General Graphs, by Chiranjib Mukherjee and Konstantin Recke View PDF HTML (experimental) 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. Comments: Minor revisions in the exposition, remarks and literature Subjects: Probability (math.PR) ; Metric Geometry (math.MG) Cite as: arXiv:2406.04222 [math.PR] (or arXiv:2406.04222v3 [math.PR] for this version) https://doi.org/10.48550/arXiv.2406.04222 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Chiranjib Mukherjee [ view email ] [v1] Thu, 6 Jun 2024 16:20:27 UTC (24 KB) [v2] Mon, 29 Jul 2024 18:01:27 UTC (26 KB) [v3] Wed, 29 Apr 2026 13:18:33 UTC (29 KB) Full-text links: Access Paper: View a PDF of the paper titled Coarse embeddability, $L^1$-compression and Percolations on General Graphs, by Chiranjib Mukherjee and Konstantin Recke View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2024-06 Change to browse by: math math.MG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from