Transportation-information inequalities for continuum Gibbs measures

Yutao Ma,Ran Wang,Liming Wu

doi:10.1214/ecp.v16-1670

Abstract

The objective of this paper is to establish explicit concentration inequalities for the Glauber dynamics related with continuum or discrete Gibbs measures. At first we establish the optimal transportation-information $W_1 I$-inequality for the $M/M/\infty$-queue associated with the Poisson measure, which improves several previous known results. Under the Dobrushin's uniqueness condition, we obtain some explicit $W_1 I$-inequalities for Gibbs measures both in the continuum and in the discrete lattice. Our method is a combination of Lipschitzian spectral gap, the Lyapunov test function approach and the tensorization technique.

Highlights

1.1 Transportation-information inequalities W1ILet be a Polish space equipped with the Borel σ-field, and let d be a lower semi-continuous metric on the product space ×
1/p d(x, y)pdπ(x, y), where the infimum is taken over all probability measures π on the product space × with marginal distributions μ and ν (say coupling of (μ, ν))
This infimum is finite once μ and ν belong to p 1 (

Summary

Transportation-information inequalities W1I

Let be a Polish space equipped with the Borel σ-field , and let d be a lower semi-continuous metric on the product space × (which does not necessarily generate the topology of ). 1/p d(x, y)pdπ(x, y) , where the infimum is taken over all probability measures π on the product space × with marginal distributions μ and ν (say coupling of (μ, ν)). This infimum is finite once μ and ν belong to p 1. Given a family of bounded measurable functions g (say g ∈ b ), the following properties are equivalent:. Gao and the third named author [6] proved a tensorization result for the Wasserstein distance (see Lemma 4.2 below) and established the “dimension-free" transportation-information inequalities Wp I(p ≥ 1) for the discrete Gibbs measure, under the Dobrushin’s uniqueness condition ([3, 4])

Continuum Gibbs measure and generator of the Glauber dynamic

W1I-inequality for continuum Gibbs measure

W1I-inequality

Findings

W1I-inequality for the discrete spin system