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

##### 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
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##### 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)
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