cea-01585492
https://cea.hal.science/cea-01585492
https://cea.hal.science/cea-01585492/document
https://cea.hal.science/cea-01585492/file/1605.06945.pdf
arxiv:1605.06945
doi:10.1088/1742-5468/2016/08/083303
[CEA] CEA - Commissariat à l'énergie atomique
[CNRS] CNRS - Centre national de la recherche scientifique
[DSM-IPHT] IPHT
[CEA-UPSAY] CEA - Université Paris-Saclay
[UNIV-PARIS-SACLAY] Université Paris-Saclay
[CEA-UPSAY-SACLAY] CEA-UPSAY-SACLAY
[CEA-DRF] Direction de Recherche Fondamentale
[ANR] ANR
[GS-MATHEMATIQUES] Graduate School Mathématiques
[GS-PHYSIQUE] Graduate School Physique
Circular Coloring of Random Graphs: Statistical Physics Investigation
Schmidt, Christian
Guenther, Nils-Eric
Zdeborová, Lenka
[PHYS.COND.CM-SM] Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]
ART
Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.
2016-05-23
2017-09-11
en
Journal of Statistical Mechanics: Theory and Experiment
IOP Publishing