The Condensation Phase Transition in Random Graph Coloring
Communications in Mathematical Physics(2015)
摘要
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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关键词
Cond,Random Graph,Tame,Isomorphism Class,Random Tree
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