Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder

Jara, M. D.; Landim, C.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 341-361.

Soit $\{{\xi }_x:x\in \mathbb{Z}\}$ une suite de variables aléatoires i.i.d. bornées supérieurement et inférieurement par des constantes finies et strictement positives. Nous étudions le théorème central limite «quenched» pour la position d’une particule marquée dans l’exclusion simple symmétrique unidimensionnelle où les variables d’occupation des sites $x$ et $x+1$ sont échangés à taux ${\xi }_x$. Nous démontrons que la position de la particule marquée converge à l’échelle diffusive vers un processus gaussien si les particules sont initiallement distribuées d’après une mesure de Bernoulli associée à un profil lisse ${\rho }_0:ℝ\to \left[0,1\right]$.

For a sequence of i.i.d. random variables $\left\{{\xi }_x:x\in \mathbb{Z}\right\}$ bounded above and below by strictly positive finite constants, consider the nearest-neighbor one-dimensional simple exclusion process in which a particle at $x(\mathrm{resp}.x+1)$ jumps to $x+1(\mathrm{resp}.x)$ at rate ${\xi }_x$. We examine a quenched non-equilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder $\{{\xi }_x:x\in \mathbb{Z}\}$. We prove that the position of the tagged particle converges under diffusive scaling to a gaussian process if the other particles are initially distributed according to a Bernoulli product measure associated to a smooth profile ${\rho }_0:ℝ\to \left[0,1\right]$.

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