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On the Wasserstein distance and the Dobrushin uniqueness theorem

Tony C. Dorlas, Baptiste Savoie

Abstract

In this paper, we revisit the Dobrushin uniqueness theorem for Gibbs measures of lattice systems of interacting particles at thermal equilibrium. In a nutshell, Dobrushin's uniqueness theorem provides a practical way to derive sufficient conditions on the inverse temperature and/or model parameters assuring uniqueness of Gibbs measures by reducing the uniqueness problem to a suitable estimate of the Wasserstein distance between pairs of 1-point Gibbs measures with different boundary conditions. After proving a general result of completeness for the Wasserstein distance, we reformulate the Dobrushin uniqueness theorem in a convenient form for lattice systems of interacting particles described by Hamiltonians that are not necessarily translation-invariant with possibly infinite-range pair-potentials and with general complete metric spaces as single-spin spaces. Our reformulation includes existence, and covers both classical lattice systems and the Euclidean version of quantum lattice systems. Further, a generalization to the Dobrushin-Shlosman theorem for translation-invariant lattice systems is given. Subsequently, we give a series of applications of these uniqueness criteria to some high-temperature classical lattice systems including the Heisenberg, Potts and Ising models. An application to classical lattice systems for which the local Gibbs measures are convex perturbations of Gaussian measures is also given.

On the Wasserstein distance and the Dobrushin uniqueness theorem

Abstract

In this paper, we revisit the Dobrushin uniqueness theorem for Gibbs measures of lattice systems of interacting particles at thermal equilibrium. In a nutshell, Dobrushin's uniqueness theorem provides a practical way to derive sufficient conditions on the inverse temperature and/or model parameters assuring uniqueness of Gibbs measures by reducing the uniqueness problem to a suitable estimate of the Wasserstein distance between pairs of 1-point Gibbs measures with different boundary conditions. After proving a general result of completeness for the Wasserstein distance, we reformulate the Dobrushin uniqueness theorem in a convenient form for lattice systems of interacting particles described by Hamiltonians that are not necessarily translation-invariant with possibly infinite-range pair-potentials and with general complete metric spaces as single-spin spaces. Our reformulation includes existence, and covers both classical lattice systems and the Euclidean version of quantum lattice systems. Further, a generalization to the Dobrushin-Shlosman theorem for translation-invariant lattice systems is given. Subsequently, we give a series of applications of these uniqueness criteria to some high-temperature classical lattice systems including the Heisenberg, Potts and Ising models. An application to classical lattice systems for which the local Gibbs measures are convex perturbations of Gaussian measures is also given.
Paper Structure (39 sections, 31 theorems, 391 equations, 4 figures, 6 tables)

This paper contains 39 sections, 31 theorems, 391 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction.
  2. Wasserstein distance and completeness.
  3. Dobrushin's uniqueness theorem revisited.
  4. Gibbs distributions.
  5. The Dobrushin uniqueness and existence theorem.
  6. Some corollaries.
  7. Proof of Theorem \ref{['Dobrushin']}.
  8. Some preliminary results.
  9. Proof of existence.
  10. Proof of uniqueness.
  11. Proof of Corollaries \ref{['corol1']} and \ref{['corol2']}.
  12. A generalization: The Dobrushin-Shlosman Theorem
  13. Applications - Part I: Uniqueness of Gibbs measures at high temperature.
  14. A general result with application to the classical Heisenberg model.
  15. The Potts and Ising models.
  16. ...and 24 more sections

Key Result

Theorem 2.1

$(i)$. The Wasserstein distance $\rho_{W}$ is a metric on $\mathcal{P}_{1}(\mathcal{X})$. $(ii)$. If $(\mathcal{X},\rho)$ is a complete metric space, then so is $(\mathcal{P}_{1}(\mathcal{X}), \rho_{W})$.

Figures (4)

  • Figure 1: Graphs of $\rho_{W}$ versus $\beta$ for $q=4$.
  • Figure 2: Graphs of $\rho_{W}$ versus $\beta$ for $q=30$.
  • Figure 3: The Dobrushin lower bound for $\beta_{c}J$ of the Potts model in dimension $d=2$ as a function of $q$.
  • Figure 4: The Dobrushin lower bound for $\beta_{c}J$ of the Potts model in dimension $d=3$ as a function of $q$.

Theorems & Definitions (54)

  • Theorem 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • Remark 3.6
  • Remark 3.7
  • Remark 3.8
  • Remark 3.9
  • ...and 44 more