TY - JOUR

T1 - Gibbsian Representations of Continuous Specifications

T2 - The Theorems of Kozlov and Sullivan Revisited

AU - Barbieri, Sebastián

AU - Gómez, Ricardo

AU - Marcus, Brian

AU - Meyerovitch, Tom

AU - Taati, Siamak

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

PY - 2021/3/1

Y1 - 2021/3/1

N2 - The theorems of Kozlov and Sullivan characterize Gibbs measures as measures with positive continuous specifications. More precisely, Kozlov showed that every positive continuous specification on symbolic configurations of the lattice is generated by a norm-summable interaction. Sullivan showed that every shift-invariant positive continuous specification is generated by a shift-invariant interaction satisfying the weaker condition of variation-summability. These results were proven in the 1970s. An open question since that time is whether Kozlov’s theorem holds in the shift-invariant setting, equivalently whether Sullivan’s conclusion can be improved from variation-summability to norm-summability. We show that the answer is no: there exist shift-invariant positive continuous specifications that are not generated by any shift-invariant norm-summable interaction. On the other hand, we give a complete proof of an extension, suggested by Kozlov, of Kozlov’s theorem to a characterization of positive continuous specifications on configuration spaces with arbitrary hard constraints. We also present an extended version of Sullivan’s theorem. Aside from simplifying some of the arguments in the original proof, our new version of Sullivan’s theorem applies in various settings not covered by the original proof. In particular, it applies when the support of the specification is the hard-core shift or the two-dimensional q-coloring shift for q≥ 6.

AB - The theorems of Kozlov and Sullivan characterize Gibbs measures as measures with positive continuous specifications. More precisely, Kozlov showed that every positive continuous specification on symbolic configurations of the lattice is generated by a norm-summable interaction. Sullivan showed that every shift-invariant positive continuous specification is generated by a shift-invariant interaction satisfying the weaker condition of variation-summability. These results were proven in the 1970s. An open question since that time is whether Kozlov’s theorem holds in the shift-invariant setting, equivalently whether Sullivan’s conclusion can be improved from variation-summability to norm-summability. We show that the answer is no: there exist shift-invariant positive continuous specifications that are not generated by any shift-invariant norm-summable interaction. On the other hand, we give a complete proof of an extension, suggested by Kozlov, of Kozlov’s theorem to a characterization of positive continuous specifications on configuration spaces with arbitrary hard constraints. We also present an extended version of Sullivan’s theorem. Aside from simplifying some of the arguments in the original proof, our new version of Sullivan’s theorem applies in various settings not covered by the original proof. In particular, it applies when the support of the specification is the hard-core shift or the two-dimensional q-coloring shift for q≥ 6.

UR - http://www.scopus.com/inward/record.url?scp=85100781418&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-03979-2

DO - 10.1007/s00220-021-03979-2

M3 - Article

AN - SCOPUS:85100781418

SN - 0010-3616

VL - 382

SP - 1111

EP - 1164

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -