The logarithmic Sobolev inequality for discrete spin systems on a lattice
The main theme of the paper is a derivation of a discrete analog of the classical logarithmic Sobolev (LS) inequality (involving a gradient \(\nabla\) being a differential operator) by using a concept of discrete gradient instead of \(\nabla\). The arguments utilize hypercontractivity estimates which are known to be equivalent to LS. Let \(\Omega=Q^ \Gamma\) be a configuration space, where \(\Gamma\) is the \(d\)-dimensional square lattice \(Z^ d\) and \(Q\) denotes a finite set called the spin space. A stochastic dynamics is introduced via a family of transition probability functions on \(\Omega\) induced by a shift invariant finite range Gibbs potential, which gives rise to a Markov semigroup of operators \(P_ t=\exp(tL)\), acting on bounded functions on \(\Omega\), with \(L\) being the generator of the semigroup. The result states that under Dobrushin-Shlosman uniqueness of the Gibbs state \(\mu\) (probability measure on \(\Omega\)) the following contractions hold: \[ ||| P_ t f||| \leq ||| f||| e^{-ct}, \qquad \| P_ t\varphi-\mu\varphi\| \leq K||| \varphi||| e^{-dt}, \] where \(c\), \(d\), \(K\) are positive constants, \(0<t<\infty\), whereas \(|||\cdot|||\) and \(\|\cdot\|\) are a seminorm involving the discrete gradient of \(f:\Omega\to R\) and sup norm for bounded functions on \(\Omega\), respectively. The authors also discuss a connection between the LS and the spectral gap SG of the generator \(L\).
scientific article; zbMATH DE number 95700
scientific article