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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
\[ \sum_{c\in X} \Vert \Phi_X \Vert\cdot (\vert X\vert - 1) < 1.\]

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
Full Text: DOI
[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)
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