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Restriction and mixing properties of interacting particle systems with unbounded range

Benedikt Jahnel, Jonas Köppl

Abstract

We consider interacting particle systems with unbounded interaction range on general countably infinite graphs $S$ and prove explicit non-asymptotic error bounds for approximations of the infinite-volume dynamics by systems of finitely many interacting particles. Moreover, we also provide non-asymptotic quantitative bounds on the spatial decay of correlations at times $t>0$ and then apply these results to show that interacting particle systems on $\mathbb{Z}$ whose interaction strengths decays exponentially fast cannot spontaneously break the time-translation symmetry, neither in the strong, nor in the weak sense.

Restriction and mixing properties of interacting particle systems with unbounded range

Abstract

We consider interacting particle systems with unbounded interaction range on general countably infinite graphs and prove explicit non-asymptotic error bounds for approximations of the infinite-volume dynamics by systems of finitely many interacting particles. Moreover, we also provide non-asymptotic quantitative bounds on the spatial decay of correlations at times and then apply these results to show that interacting particle systems on whose interaction strengths decays exponentially fast cannot spontaneously break the time-translation symmetry, neither in the strong, nor in the weak sense.
Paper Structure (18 sections, 14 theorems, 97 equations)

This paper contains 18 sections, 14 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Interacting particle system
  4. Main results
  5. Restriction to finite volumes
  6. Approximation of stationary measures
  7. Spatial decay of correlations
  8. Quantitative decay of correlations for limiting stationary measures
  9. Absence of time-translation symmetry breaking in one dimension
  10. Proof strategy for the absence of time-translation symmetry breaking
  11. Restriction property and decay of correlations
  12. Propagation of bounds
  13. Restriction to finite volumes
  14. Approximation of stationary measures
  15. Spatial decay of correlations
  16. ...and 3 more sections

Key Result

Theorem 2.1

Assume that the conditions $\mathbf{(L1)}$ and $\mathbf{(R1)-(R4)}$ are satisfied. Let $\Lambda \Subset S$ and $h>0$. Then, for all $\Lambda$-local functions $f\colon\Omega \to \mathbb{R}$ we have where the constant $\mathbf{C}_{S,L}$ is a geometric quantity defined by In particular, for any initial distribution $\mu \in \mathcal{M}_1(\Omega)$ the total variation error in $\Lambda$ is bounded by

Theorems & Definitions (25)

  • Theorem 2.1: Restriction to finite volumes
  • Theorem 2.2: Approximation of stationary measures
  • Theorem 2.3: Spatial decay of correlations
  • Theorem 2.4: Spatial mixing for limiting stationary measures
  • Theorem 2.5: Absence of time-translation symmetry breaking
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 15 more