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Weakly asymmetric facilitated exclusion process

Guillaume Barraquand, Oriane Blondel, Marielle Simon

TL;DR

The paper analyzes fluctuations in the weakly asymmetric facilitated exclusion process (FEP) on $\mathbb Z$ by mapping to ASEP on the half-line with a boundary reservoir and performing a Hopf-Cole transform to obtain a discrete stochastic heat equation (SHE). In the weak-asymmetry limit, with microscopic scale $p=\tfrac12 e^{\varepsilon}$, $q=\tfrac12 e^{-\varepsilon}$, the rescaled Hopf-Cole field converges to the SHE on $\mathbb R_+$ with Dirichlet boundary, starting from an initial condition given by the derivative of a Dirac delta, $-2\delta_0'$. The main results cover both empty and near-equilibrium initial conditions, with a detailed construction of the macroscopic solution and a rigorous proof of convergence, leveraging discrete-to-continuum martingale problems, precise heat-kernel bounds, and Markov duality for moments. The work provides a rigorous link between microscopic FEP dynamics and half-space KPZ universality, including novel boundary-killing estimates and the delta-prime initial data, which broadens understanding of KPZ fluctuations in domains with boundaries.

Abstract

We consider the facilitated exclusion process, an interacting particle system on the integer line where particles hop to one of their left or right neighbouring site only when the other neighbouring site is occupied by a particle. A peculiarity of this system is that, starting from the step initial condition, the density profile develops a downward jump discontinuity around the position of the first particle, unlike other exclusion processes such as the asymmetric simple exclusion process (ASEP). In the weakly asymmetric regime, we show that the field of particle positions around the jump discontinuity converges to the solution of the multiplicative noise stochastic heat equation (i.e. the exponential of a solution to the KPZ equation) on a half-line subject to Dirichlet boundary condition, with initial condition given by the derivative of a Dirac delta function. We prove this result by reformulating the problem in terms of ASEP on a half-line with a boundary reservoir, for which we extend known proofs of convergence to deal with Dirichlet boundary condition and the very singular type of initial condition that arises in our case.

Weakly asymmetric facilitated exclusion process

TL;DR

The paper analyzes fluctuations in the weakly asymmetric facilitated exclusion process (FEP) on by mapping to ASEP on the half-line with a boundary reservoir and performing a Hopf-Cole transform to obtain a discrete stochastic heat equation (SHE). In the weak-asymmetry limit, with microscopic scale , , the rescaled Hopf-Cole field converges to the SHE on with Dirichlet boundary, starting from an initial condition given by the derivative of a Dirac delta, . The main results cover both empty and near-equilibrium initial conditions, with a detailed construction of the macroscopic solution and a rigorous proof of convergence, leveraging discrete-to-continuum martingale problems, precise heat-kernel bounds, and Markov duality for moments. The work provides a rigorous link between microscopic FEP dynamics and half-space KPZ universality, including novel boundary-killing estimates and the delta-prime initial data, which broadens understanding of KPZ fluctuations in domains with boundaries.

Abstract

We consider the facilitated exclusion process, an interacting particle system on the integer line where particles hop to one of their left or right neighbouring site only when the other neighbouring site is occupied by a particle. A peculiarity of this system is that, starting from the step initial condition, the density profile develops a downward jump discontinuity around the position of the first particle, unlike other exclusion processes such as the asymmetric simple exclusion process (ASEP). In the weakly asymmetric regime, we show that the field of particle positions around the jump discontinuity converges to the solution of the multiplicative noise stochastic heat equation (i.e. the exponential of a solution to the KPZ equation) on a half-line subject to Dirichlet boundary condition, with initial condition given by the derivative of a Dirac delta function. We prove this result by reformulating the problem in terms of ASEP on a half-line with a boundary reservoir, for which we extend known proofs of convergence to deal with Dirichlet boundary condition and the very singular type of initial condition that arises in our case.
Paper Structure (38 sections, 27 theorems, 241 equations, 1 figure)

This paper contains 38 sections, 27 theorems, 241 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Facilitated exclusion process
  3. KPZ equation and Hopf-Cole transform
  4. KPZ equation on the positive half-line
  5. Main results
  6. Comparison with previous literature on half-space KPZ equation
  7. Outline of the paper
  8. Microscopic models and mapping
  9. Main results and strategy of the proof
  10. Hopf-Cole transform
  11. Weak asymmetry
  12. Main theorems
  13. Empty initial condition
  14. Near-equilibrium initial condition
  15. An example of near-equilibrium initial condition
  16. ...and 23 more sections

Key Result

Lemma 2.1

Assume that the generator of the Markov process $\{\eta_t\}_{t\geqslant 0}$ is $\mathcal{L}_F$ given in eq:L, then $\{\mathfrak{S}(\eta_t)\}_{t\geqslant 0}$ is an asymmetric simple exclusion process on the infinite half-line $\mathbb N^*$ with a boundary reservoir which injects particles at rate $p$ where $\sigma^1$ is obtained after the creation of a particle at site 1, i.e.$\sigma^1(k)=\mathbf{1

Figures (1)

  • Figure 1: The top figure represents a configuration $\eta$ in $\overline\Omega$ and the possible transitions with their respective rates. The bottom figure represents the mapped configuration $\mathfrak{S}(\eta)$ on $\lbrace 0,1\rbrace^{\mathbb N^*}$ and the possible transitions, in the same color as the corresponding transitions in $\eta$.

Theorems & Definitions (59)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Remark 3.4
  • Proposition 3.5
  • ...and 49 more