Estimates of logarithmic Sobolev constat for finite-volume continuous spin systems

Abstract

LetM be a compact, connected Riemannian manifold (with or without boundary); we study the logarithmic Sobolev constant for stochastic Ising models on\(M^{Z^d } \). Let {λ} be a sequence of cubes inZ d ; we show that the logarithmic Sobolev constant for the finite systems onM A shrinks at most exponentially fast in |Δ|(d-1)/d (d≥2), which is sharp in order for the classical Ising models withM=[−1, 1]. Moreover, a geometrical lemma proved by L. E. Thomas is also improved.