TY - JOUR
T1 - Rigidity of proper colorings of Zd
AU - Peled, Ron
AU - Spinka, Yinon
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023/4
Y1 - 2023/4
N2 - A proper q-coloring of a domain in Zd is a function assigning one of q colors to each vertex of the domain such that adjacent vertices are colored differently. Sampling a proper q-coloring uniformly at random, does the coloring typically exhibit long-range order? It has been known since the work of Dobrushin that no such ordering can arise when q is large compared with d. We prove here that long-range order does arise for each q when d is sufficiently high, and further characterize all periodic maximal-entropy Gibbs states for the model. Ordering is also shown to emerge in low dimensions if the lattice Zd is replaced by Zd1×Td2 with d1≥ 2 , d= d1+ d2 sufficiently high and T a cycle of even length. The results address questions going back to Berker and Kadanoff (in J Phys A Math Gen 13(7):L259, 1980), Kotecký (in Phys Rev B 31(5):3088, 1985) and Salas and Sokal (in J Stat Phys 86(3):551–579, 1997).
AB - A proper q-coloring of a domain in Zd is a function assigning one of q colors to each vertex of the domain such that adjacent vertices are colored differently. Sampling a proper q-coloring uniformly at random, does the coloring typically exhibit long-range order? It has been known since the work of Dobrushin that no such ordering can arise when q is large compared with d. We prove here that long-range order does arise for each q when d is sufficiently high, and further characterize all periodic maximal-entropy Gibbs states for the model. Ordering is also shown to emerge in low dimensions if the lattice Zd is replaced by Zd1×Td2 with d1≥ 2 , d= d1+ d2 sufficiently high and T a cycle of even length. The results address questions going back to Berker and Kadanoff (in J Phys A Math Gen 13(7):L259, 1980), Kotecký (in Phys Rev B 31(5):3088, 1985) and Salas and Sokal (in J Stat Phys 86(3):551–579, 1997).
UR - http://www.scopus.com/inward/record.url?scp=85142717602&partnerID=8YFLogxK
U2 - 10.1007/s00222-022-01164-3
DO - 10.1007/s00222-022-01164-3
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:85142717602
SN - 0020-9910
VL - 232
SP - 79
EP - 162
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 1
ER -