[Paper Review] Poincar\\'e, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

Matthias Erbar, Max Fathi|arXiv (Cornell University)|Dec 1, 2016

Geometric Analysis and Curvature Flows28 references3 citations

TL;DR

This paper establishes discrete analogues of classical Riemannian geometric inequalities—Poincaré, modified logarithmic Sobolev, and isoperimetric—under non-negative entropic Ricci curvature for finite Markov chains. Using a Wasserstein-type transport distance $\mathcal{W}$, it proves that spectral gap, Cheeger constant, and modified logarithmic Sobolev constant are bounded from below by constants depending only on the diameter $D$ of the state space, extending Li-Yau, Buser, and Wang's results to discrete settings with non-positive curvature bounds.

ABSTRACT

We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang.

Motivation & Objective

• To extend classical functional inequalities from Riemannian geometry to discrete Markov chains under non-negative entropic Ricci curvature.
• To establish quantitative lower bounds on the spectral gap, Cheeger constant, and modified logarithmic Sobolev constant for such chains.
• To show that these bounds depend only on the diameter $D$ of the state space under the $\mathcal{W}$-distance, not on the dimension or other structural details.
• To bridge the gap between continuous curvature-based geometric analysis and discrete probability models, particularly for zero-range processes and random walks on graphs.

Proposed method

• The authors define a discrete transport distance $\mathcal{W}$ on probability measures, which induces a metric $d_{\mathcal{W}}$ on the state space $\mathcal{X}$, replacing the $L^2$-Wasserstein distance used in Lott-Sturm-Villani theory.
• They use the entropic Ricci curvature notion from [29, 13], which involves convexity of entropy along $\mathcal{W}$-geodesics, and define curvature lower bounds via the $\Gamma$-calculus framework.
• The key technical tool is a convexity inequality for the entropy along $\mathcal{W}$-geodesics: $\mathcal{H}(\rho_{1/2}) \leq \frac{1}{2}\mathcal{H}(\delta_x) + \frac{1}{2}\mathcal{H}(\delta_y) - \frac{\kappa}{8}d_{\mathcal{W}}(x,y)^2$, which links curvature to distance.
• They derive diameter bounds via a discrete Bonnet-Myers theorem: if $\operatorname{Ric} \geq \kappa > 0$, then $d_{\mathcal{W}}(x,y) \leq 2\sqrt{(-\log \pi(x) - \log \pi(y))/\kappa}$, leading to diameter control in terms of $\pi_*$.
• For the main results, they apply triangle inequality and lifting techniques to bound $\mathcal{W}(\delta_\eta, \delta_{\tilde{\eta}})$ for configurations differing by one particle, reducing to the complete graph case with rate $1/L$, and use known mixing time estimates.

Experimental results

Research questions

• RQ1 Can functional inequalities such as the Poincaré, modified logarithmic Sobolev, and isoperimetric inequalities be established for discrete Markov chains under non-negative entropic Ricci curvature?
• RQ2 What is the dependence of the spectral gap and modified logarithmic Sobolev constant on the diameter $D$ of the state space under the $\mathcal{W}$-metric?
• RQ3 How do these discrete inequalities compare to their continuous Riemannian counterparts (e.g., Li-Yau, Buser, Wang) in terms of sharpness and asymptotic behavior?
• RQ4 Can the diameter of the state space under $d_{\mathcal{W}}$ be bounded from above using curvature and stationary measure, even in high-dimensional or non-uniform settings?
• RQ5 What is the asymptotic behavior of the modified logarithmic Sobolev constant for the zero-range process with fixed particle density?

Key findings

• The Cheeger constant $h$ satisfies $h \geq \frac{1}{3}\sqrt{Q_* \lambda_1}$, where $Q_*$ is the minimal transition rate, establishing a discrete analogue of Buser's inequality.
• Under non-negative entropic Ricci curvature and diameter $D$ under $d_{\mathcal{W}}$, the spectral gap satisfies $\lambda_1 \geq \frac{c}{D^2}$ for a universal constant $c$, generalizing Li-Yau's result.
• The modified logarithmic Sobolev constant is bounded from below by $\frac{c'}{D^2}$ for another universal constant $c'$, extending Wang's inequality to the discrete setting.
• For the zero-range process on $L$ sites with $K$ particles, the diameter under $d_{\mathcal{W}}$ is bounded by $cK\sqrt{L\log L}$, implying that the mLSI constant behaves like $1/(K^2 L \log L)$ at fixed density $K/L$, suggesting a $1/L^2$ scaling.
• A discrete Bonnet-Myers theorem holds: if $\operatorname{Ric} \geq \kappa > 0$, then $d_{\mathcal{W}}(x,y) \leq 2\sqrt{(-\log \pi(x) - \log \pi(y))/\kappa}$, with diameter bounded by $2\sqrt{-2\log \pi_* / \kappa}$.