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The Condensation Phase Transition in Random Graph Coloring

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Abstract

Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry of interactions is induced by a sparse random graph or hypergraph. One example of such a model is the graph coloring problem on the Erdős–Renyi random graph G(n, d/n), which can be viewed as the zero temperature case of the Potts antiferromagnet. The cavity method predicts that in addition to the k-colorability phase transition studied intensively in combinatorics, there exists a second phase transition called the condensation phase transition (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007). In fact, there is a conjecture as to the precise location of this phase transition in terms of a certain distributional fixed point problem. In this paper we prove this conjecture for k exceeding a certain constant k 0.

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Authors and Affiliations

  1. Mathematics Institute, Goethe University, 10 Robert Mayer St, 60325, Frankfurt, Germany

    Victor Bapst, Amin Coja-Oghlan, Samuel Hetterich & Felicia Raßmann

  2. Department of Communication Systems Engineering, Ben-Gurion University of the Negev, Beer-sheba, Israel

    Dan Vilenchik

Authors
  1. Victor Bapst
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  2. Amin Coja-Oghlan
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  3. Samuel Hetterich
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  4. Felicia Raßmann
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  5. Dan Vilenchik
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Corresponding author

Correspondence to Amin Coja-Oghlan.

Additional information

Communicated by H.-T. Yau

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 278857-PTCC.

An extended abstract version of this work appeared in the Proceedings of the 18th International Workshop on Randomization and Computation (‘RANDOM’) 2014.

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Bapst, V., Coja-Oghlan, A., Hetterich, S. et al. The Condensation Phase Transition in Random Graph Coloring. Commun. Math. Phys. 341, 543–606 (2016). https://doi.org/10.1007/s00220-015-2464-z

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  • DOI: https://doi.org/10.1007/s00220-015-2464-z

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