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 -