×

A remark on Dobrushin’s uniqueness theorem. (English) Zbl 0435.60099

It is shown that for a lattice system with compact one-point state space and translation invariant interaction, whose “formal Hamiltonian” is given by \(H(s) = \sum_X \Phi_X(s)\), \(X\) running through all finite subsets of \(\mathbb Z^d\), the hypothesis of Dobrushin’s theorem on the uniqueness of Gibbs measures – see R. L. Dobrushin [Teor. Veroyatn. Primen. 13, 201–229 (1968; Zbl 0177.45202)] is satisfied if
\[ \sum_{c\in X} \Vert \Phi_X \Vert\cdot (\vert X\vert - 1) < 1.\]

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B05 Classical equilibrium statistical mechanics (general)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Citations:

Zbl 0177.45202
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dobrushin, R.L.: Theory Probab. Its Appl.13, 197-224 (1968) · doi:10.1137/1113026
[2] Griffiths, R.: Commun. Math. Phys.6, 121-127 (1967) · doi:10.1007/BF01654128
[3] Gross, L.: Preprint, Cornell University (1979)
[4] Israel, R.: Commun. Math. Phys.50, 245-257 (1976) · doi:10.1007/BF01609405
[5] Israel, R.: Commun. Math. Phys.43, 59-68 (1975) · Zbl 0309.46056 · doi:10.1007/BF01609141
[6] Lanford, O. III: Lecture notes in physics, Vol.20. Berlin, Heidelberg, New York: Springer 1973
[7] Vaserstein, L.: Prob. Trans. Infor.5, No. 3, 64-72 (1969)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.