The condensation phase transition in random graph coloring

Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random $k$-SAT or random graph $k$-coloring, very shortly before the...

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Veröffentlicht in:	arXiv.org 2014-04
Hauptverfasser:	Bapst, Victor, Amin Coja-Oghlan, Hetterich, Samuel, Rassmann, Felicia, Vilenchik, Dan
Format:	Artikel
Sprache:	eng
Schlagworte:	Algorithms Computer Science - Discrete Mathematics Condensation Graph coloring Mathematics - Combinatorics Mathematics - Probability Message passing Phase transitions Physicists
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description	Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random $k$-SAT or random graph $k$-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called "condensation" [Krzakala et al., PNAS 2007]. The existence of this phase transition appears to be intimately related to the difficulty of proving precise results on, e.g., the $k$-colorability threshold as well as to the performance of message passing algorithms. In random graph $k$-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture for $k$ exceeding a certain constant $k_0$.
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subjects	Algorithms Computer Science - Discrete Mathematics Condensation Graph coloring Mathematics - Combinatorics Mathematics - Probability Message passing Phase transitions Physicists
title	The condensation phase transition in random graph coloring
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