The log-Sobolev inequality for spin systems of higher order interactions

Abstract

We study the infinite-dimensional log-Sobolev inequality for spin systems on ${\mathbb{Z}}^d$ with interactions of power higher than quadratic. We assume that the one site measure without a boundary $e^{-\varphi (x)}dx/Z$ satisfies a log-Sobolev inequality and we determine conditions so that the infinite-dimensional Gibbs measure also satisfies the inequality. As a concrete application, we prove that a certain class of nontrivial Gibbs measures with non-quadratic interaction potentials on an infinite product of Heisenberg groups satisfy the log-Sobolev inequality.

References in corpus (17)

• Characterization of Talagrand's transport-entropy inequalities in metric spaces
• Logarithmic Sobolev inequalities for infinite dimensional Hörmander type generators on the Heisenberg group
• On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group
• Sharp log-Sobolev inequalities in $\mathsf{CD}(0,N)$ spaces with applications
• Entropy dissipation for degenerate stochastic differential equations via sub-Riemannian density manifold
• The Logarithmic Sobolev Inequality in Infinite dimensions for Unbounded Spin Systems on the Lattice with non Quadratic Interactions
• Note on the $q$-Logarithmic Sobolev and $p$-Talagrand Inequalities on Carnot Groups
• A note on the Modified Log-Sobolev inequality
• The Logarithmic Sobolev Inequality for Gibbs measures on infinite product of Heisenberg groups
• Sub-Riemannian Ricci curvature via generalized Gamma $z$ calculus
• Hypoelliptic entropy dissipation for stochastic differential equations
• Concentration inequalities for Gibbs measures
• Logarithmic Schrödinger Equations in Infinite Dimensions
• Some super-Poincaré inequalities for gaussian-like measures on stratified Lie groups
• The log-Sobolev inequality with quadratic interactions
• Log-Sobolev inequalities for infinite-dimensional Gibbs measures with non-quadratic interactions
• Spectral Gap Inequalities on Nilpotent Lie Groups in Infinite Dimensions