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Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems. (English) Zbl 1022.60087
For a system of unbounded spins over the $$d$$-dimensional cube $$\{1,\ldots,L\}^d$$, $$L\geq 1$$, the paper considers evolutions governed by reversible conservative Ginzburg-Landau processes having Gaussian potential perturbed additively by bounded smooth functions $$F:\mathbb{R}\rightarrow\mathbb{R}$$ with bounded derivative. The upper bounds of the form $$cL^2$$, $$c$$ being a constant depending on $$F$$ and independent of $$d$$, are established for the inverses of the spectral gap of the generators and of the constants in logarithmic Sobolev inequalities related to the processes (in the latter case $$||F''||_{\infty}<\infty$$ is required). Following the approach developed by S. L. Lu and H. T. Yau [Commun. Math. Phys. 156, 399-433 (1993; Zbl 0779.60078)], the authors overcome the technical difficulties due to unboundedness of spins. The main ingredients of the proof are a local CLT uniform over the parameter and some sharp large deviations estimates.

##### 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 60F05 Central limit and other weak theorems
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