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Regularity of Gibbs measures for unbounded spin systems on general graphs

Christoforos Panagiotis, William Veitch

Abstract

We consider a general class of spin systems with potentially unbounded real-valued spins, defined via a single-site potential with super-Gaussian tails on general graphs, allowing for both short- and long-range interactions. This class includes all $P(\varphi)$ models, in particular the well-studied $\varphi^4$ model. We construct an infinite-volume extremal measure called the plus measure as the limit of finite-volume Gibbs measures with weakly growing boundary conditions and show that it is regular, in the sense that it admits a bounded Radon-Nikodym derivative with respect to a product measure of single-site distributions with super-Gaussian tails. Moreover, we provide an alternative construction of the plus measure as the limit of finite-volume Gibbs measures that are regular up to the boundary. As a key intermediate step, we establish regularity and tightness of finite-volume Gibbs measures for a large class of growing boundary conditions $ξ$. Our regularity estimates are encoded in terms of a function $A(ξ)$, which provides precise control on the change of measure induced by boundary perturbations, and can thus be viewed as an analogue of the Cameron-Martin theorem for non-Gaussian fields. In the nearest-neighbour case, this class includes boundary conditions that grow at most double-exponentially in the distance to the boundary when the single-site measure has tails of the form $e^{-a|u|^n}$ for some $n>2$.Our results apply to arbitrary graphs and improve upon earlier results of Lebowitz and Presutti, and Ruelle, which apply in the context of $\mathbb{Z}^d$ and allow only logarithmically growing boundary conditions, as well as subsequent extensions to vertex-transitive graphs of polynomial growth.

Regularity of Gibbs measures for unbounded spin systems on general graphs

Abstract

We consider a general class of spin systems with potentially unbounded real-valued spins, defined via a single-site potential with super-Gaussian tails on general graphs, allowing for both short- and long-range interactions. This class includes all models, in particular the well-studied model. We construct an infinite-volume extremal measure called the plus measure as the limit of finite-volume Gibbs measures with weakly growing boundary conditions and show that it is regular, in the sense that it admits a bounded Radon-Nikodym derivative with respect to a product measure of single-site distributions with super-Gaussian tails. Moreover, we provide an alternative construction of the plus measure as the limit of finite-volume Gibbs measures that are regular up to the boundary. As a key intermediate step, we establish regularity and tightness of finite-volume Gibbs measures for a large class of growing boundary conditions . Our regularity estimates are encoded in terms of a function , which provides precise control on the change of measure induced by boundary perturbations, and can thus be viewed as an analogue of the Cameron-Martin theorem for non-Gaussian fields. In the nearest-neighbour case, this class includes boundary conditions that grow at most double-exponentially in the distance to the boundary when the single-site measure has tails of the form for some .Our results apply to arbitrary graphs and improve upon earlier results of Lebowitz and Presutti, and Ruelle, which apply in the context of and allow only logarithmically growing boundary conditions, as well as subsequent extensions to vertex-transitive graphs of polynomial growth.

Paper Structure

This paper contains 14 sections, 23 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Paper organisation
  3. Acknowledgements
  4. Definitions and preliminaries
  5. Definition of the model
  6. Definition of $A(x,\Lambda)$
  7. Branching processes
  8. Proof of Theorem \ref{['main theorem']}
  9. Regularity for related models
  10. Corollaries and applications
  11. Results for nearest-neighbour interactions
  12. Results for more general interactions
  13. Results for infinite-volume measures
  14. An alternative construction of the plus measure

Key Result

Theorem 1.1

Let $V$ be a countable set and $(J_{x, y})_{x, y \in V}$ be admissible interactions on $V$. Let $\beta \geq 0$, $a > 0$, $n > 2$ and let $\rho$ be a single-site measure satisfying eq: single-site assumption. There exist constants $C \geq 1, \tilde{C} > 0$ such that for any $\Lambda \subset V$ finite where $\rho_{\frac{a}{2}}$ is defined by $\mathrm{d}\rho_{\frac{a}{2}}(u) = e^{\frac{a}{2} |u|^{n}}

Theorems & Definitions (53)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 43 more