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Moderate deviations for the facilitated exclusion process in equilibrium

Linjie Zhao

TL;DR

This work develops moderate deviation principles for the density fluctuation fields of the one-dimensional facilitated exclusion process (FEP) started from the stationary measure, in both symmetric diffusive and asymmetric transport regimes. The authors introduce a super-exponential Boltzmann-Gibbs principle based on a logarithmic Sobolev inequality and ensemble equivalence to control non-conserved quantities, and they derive both upper and lower bounds via exponential martingales and hydrodynamic limits. The main contributions include explicit rate functions $Q^{\rm sym}$ and $Q^{\rm asym}$, handling of non-product invariant measures, and the result that in the asymmetric case deviations of the dynamics vanish under hyperbolic scaling unless one speeds up the time scale, revealing distinct fluctuation structures. The techniques developed here have potential applicability to other interacting particle systems with LS inequalities and non-product invariant measures, offering a pathway to analyze moderate deviations beyond product-structured models.

Abstract

We derive the moderate deviation principles for the fluctuation fields of the facilitated exclusion process (FEP) in one dimension when the process starts from its stationary measure, both in the symmetric and asymmetric cases. The main step is to prove a super-exponential version of the Boltzmann-Gibbs principle, which relies on the logarithmic Sobolev inequality for the FEP.

Moderate deviations for the facilitated exclusion process in equilibrium

TL;DR

This work develops moderate deviation principles for the density fluctuation fields of the one-dimensional facilitated exclusion process (FEP) started from the stationary measure, in both symmetric diffusive and asymmetric transport regimes. The authors introduce a super-exponential Boltzmann-Gibbs principle based on a logarithmic Sobolev inequality and ensemble equivalence to control non-conserved quantities, and they derive both upper and lower bounds via exponential martingales and hydrodynamic limits. The main contributions include explicit rate functions and , handling of non-product invariant measures, and the result that in the asymmetric case deviations of the dynamics vanish under hyperbolic scaling unless one speeds up the time scale, revealing distinct fluctuation structures. The techniques developed here have potential applicability to other interacting particle systems with LS inequalities and non-product invariant measures, offering a pathway to analyze moderate deviations beyond product-structured models.

Abstract

We derive the moderate deviation principles for the fluctuation fields of the facilitated exclusion process (FEP) in one dimension when the process starts from its stationary measure, both in the symmetric and asymmetric cases. The main step is to prove a super-exponential version of the Boltzmann-Gibbs principle, which relies on the logarithmic Sobolev inequality for the FEP.
Paper Structure (24 sections, 10 theorems, 179 equations)

This paper contains 24 sections, 10 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Model and Results
  4. The Facilitated Exclusion Process
  5. Invariant measures of the FEP
  6. Moderate deviations for the fluctuation fields
  7. The symmetric case: upper bound
  8. Upper bound over compact sets
  9. Exponential tightness
  10. The martingale term
  11. The time integral term
  12. Super-exponential Boltzmann-Gibbs principle
  13. Proof of \ref{['superexp 1']}
  14. Proof of \ref{['superexp 2']}
  15. MDP for the measure $\pi_\rho$
  16. ...and 9 more sections

Key Result

Theorem 2.1

The sequence of processes $\{\mu^N_t, 0 \leqslant t \leqslant T\}$ satisfies moderate deviation principles with decay rate $a_N^2 / N$ and with rate function $Q^{\rm sym}$ in the symmetric case, and with rate function $Q^{asym}$ in the asymmetric case under the additional assumption $a_N \gg \sqrt{N where $Q=Q^{\rm sym}$ (respectively $Q=Q^{\rm asym}$) in the symmetric (respectively asymmetric) ca

Theorems & Definitions (14)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • Lemma 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 4.3
  • ...and 4 more