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The equivalence of the logarithmic Sobolev inequality and the Dobrushin- Shlosman mixing condition. (English) Zbl 0745.60104

Summary: Given a finite range lattice gas with a compact, continuous spin space, it is shown that a uniform logarithmic Sobolev inequality holds if and only if the Dobrushin-Shlosman mixing condition holds. As a consequence of our considerations, we also show that these conditions are equivalent to a statement about the uniform rate at which the associated Glauber dynamics tends to equilibrium. In this same direction, we show that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the Glauber dynamics.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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