[Paper Review] The Logarithmic Sobolev Inequality for Gibbs measures on infinite product of Heisenberg groups

Ioannis Papageorgiou|arXiv (Cornell University)|Jan 11, 2009

Mathematical Dynamics and Fractals3 citations

TL;DR

This paper establishes conditions under which the infinite-dimensional Gibbs measure on a product of Heisenberg groups satisfies the $q$-Log-Sobolev inequality, extending prior results from real-valued spins to non-Abelian Heisenberg group spins. It proves that if the one-site boundary-free measure satisfies either a $q$-Log-Sobolev or a non-uniform U-bound inequality, then the infinite-volume Gibbs measure inherits the $q$-Log-Sobolev inequality under suitable interaction conditions, relaxing prior uniform boundedness assumptions on second derivatives of interactions.

ABSTRACT

We are interested in the $q$ Logarithmic Sobolev inequality for probability measures on the infinite product of Heisenberg groups. We assume that the one site boundary free measure satisfies either a $q$ Log-Sobolev inequality or a U-Bound inequality, and we determine conditions so that the infinite dimensional Gibbs measure satisfies a $q$ Log-Sobolev inequality.

Motivation & Objective

• To extend the $q$-Log-Sobolev inequality from finite to infinite-dimensional Gibbs measures on products of Heisenberg groups.
• To analyze systems of spins valued in the Heisenberg group, a non-Abelian, sub-Riemannian Lie group, where standard elliptic methods fail.
• To relax the uniform boundedness condition on second derivatives of interactions, which previously restricted applicability in prior works.
• To establish sufficient conditions on one-site measures (either satisfying $q$-LS or a non-uniform U-bound) for the infinite-volume Gibbs measure to inherit the $q$-LS inequality.
• To generalize results from real-valued spins and earlier Heisenberg group works by handling non-uniform and singular interactions.

Proposed method

• Uses the sub-gradient $\nabla = (X_1, X_2)$ and sub-Laplacian $\Delta = X_1^2 + X_2^2$ on the Heisenberg group $\mathbb{H}$, which satisfy the Hörmander condition.
• Applies the $q$-Log-Sobolev inequality in the form $\mu(|f|^q \log \frac{|f|^q}{\mu|f|^q}) \leq c \mu(|\nabla f|^q)$ for $q \in (1,2]$.
• Employs a block Glauber dynamics approach via the projection operators $\mathcal{P}^n$ and $\mathcal{Q}^n$, which alternate between different site projections.
• Uses conditional expectations $\mathbb{E}^{\Gamma_i}$ and $\mathbb{E}^{\Gamma_j}$ to decompose the variance and control the $L^q$ norm of differences between successive projections.
• Applies Chebyshev’s inequality and the Borel-Cantelli lemma to show almost sure convergence of the sequence $\mathcal{Q}^n f$ to $\nu f$, establishing ergodicity and spectral gap-like control.
• Relies on Lemma 6.5 and Lemma 7.1 to bound the $L^q$ norms of gradients of conditional expectations, leading to exponential decay in the $q$-norm of the difference between successive projections.

Experimental results

Research questions

• RQ1 Under what conditions does the infinite-dimensional Gibbs measure on a product of Heisenberg groups satisfy the $q$-Log-Sobolev inequality?
• RQ2 Can the $q$-Log-Sobolev inequality be extended to non-Abelian spin systems on the Heisenberg group when the one-site measure satisfies a $q$-LS or a non-uniform U-bound?
• RQ3 How can the standard uniform bound on $\|\nabla_i \nabla_j V(x_i, x_j)\|_\infty$ be relaxed in the context of $q$-Log-Sobolev inequalities for Heisenberg group spins?
• RQ4 What role does the Carnot-Carathéodory distance play in characterizing the geometry of the Heisenberg group for functional inequalities?
• RQ5 Can the convergence of the Glauber dynamics $\mathcal{Q}^n f$ to the mean $\nu f$ be used to infer the $q$-Log-Sobolev inequality for the infinite-volume measure?

Key findings

• The infinite-dimensional Gibbs measure satisfies the $q$-Log-Sobolev inequality if the one-site boundary-free measure satisfies the $q$-Log-Sobolev inequality and the interaction satisfies condition (C1), which controls the growth of the gradient of the potential.
• The paper establishes the $q$-Log-Sobolev inequality for Gibbs measures even when $\|\nabla_i \nabla_j V(x_i, x_j)\|_\infty = \infty$, thus generalizing beyond the classical bounded second derivative assumption.
• The proof relies on a block dynamics argument using alternating projections $\mathcal{P}^n$ and $\mathcal{Q}^n$, which leads to exponential decay in the $L^q$ norm of the difference between successive iterates.
• The Borel-Cantelli lemma is applied to show that $\mathcal{Q}^n f \to \nu f$ almost surely, which implies the existence of a spectral gap and supports the $q$-LS inequality.
• The criterion in Theorem 2.3 applies to local specifications with no quadratic interactions, where the second derivative of the potential is unbounded, thus extending results from [22, 23, 28, 38].
• The method avoids reliance on the $\Gamma_2$-criterion, which fails on the Heisenberg group due to non-ellipticity, and instead uses a conditional expectation and gradient norm control strategy.