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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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