Abstract
Reconstruction problems have been studied in a number of contexts including biology, information theory and statistical physics. We consider the reconstruction problem for random k-colourings on the Δ-ary tree for large k. Bhatnagar et al. [2] showed non-reconstruction when \({\Delta \leq \frac12 k\log k - o(k\log k)}\) . We tighten this result and show non-reconstruction when \({\Delta \leq k[\log k + \log \log k + 1 - \log 2 -o(1)]}\) , which is very close to the best known bound establishing reconstruction which is \({\Delta \geq k[\log k + \log \log k + 1+o(1)]}\).
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Acknowledgments
The author would like to thank Elchanan Mossel for his useful comments and advice and thank Dror Weitz, Nayantara Bhatnagar, Lenka Zdeborova, Florent Krza̧kała, Guilhem Semerjian and Dmitry Panchenko for useful discussions. He would also like to thank the anonymous referees and associate editor for their careful reading of the paper and suggested improvements in the exposition.
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Communicated by F. Toninelli
Supported by NSF grants DMS-0528488 and DMS-0548249.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Sly, A. Reconstruction of Random Colourings. Commun. Math. Phys. 288, 943–961 (2009). https://doi.org/10.1007/s00220-009-0783-7
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DOI: https://doi.org/10.1007/s00220-009-0783-7