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


Zbl 0177.45202
Full Text: DOI


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