TY - JOUR
T1 - The Condensation Phase Transition in Random Graph Coloring
AU - Bapst, Victor
AU - Coja-Oghlan, Amin
AU - Hetterich, Samuel
AU - Raßmann, Felicia
AU - Vilenchik, Dan
N1 - Funding Information:
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.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.
PY - 2016/1/1
Y1 - 2016/1/1
N2 - 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 k0.
AB - 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 k0.
UR - http://www.scopus.com/inward/record.url?scp=84953359894&partnerID=8YFLogxK
U2 - 10.1007/s00220-015-2464-z
DO - 10.1007/s00220-015-2464-z
M3 - Article
AN - SCOPUS:84953359894
SN - 0010-3616
VL - 341
SP - 543
EP - 606
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 2
ER -