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), p. 341-361 / Harvested from Numdam

Résumé

Soit $\lbrace \xi _x:x\in \mathbb {Z}\rbrace $ 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 $\rho _0:\mathbb {R}\rightarrow \left[ 0,1 \right]$.

Abstract

For a sequence of i.i.d. random variables $\left\lbrace \xi _x:x\in \mathbb {Z} \right\rbrace $ 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 $\lbrace \xi _x:x\in \mathbb {Z}\rbrace $. 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 $\rho _0:\mathbb {R}\rightarrow \left[ 0,1 \right]$.

Détail

Publié le : 2008-01-01

MR 2446327 | Zbl 1195.60124 | 2 citations dans Numdam

DOI : https://doi.org/10.1214/07-AIHP112
Classification:  60K35

Bibliographie

[1] A. Faggionato. Bulk diffusion of 1D exclusion process with bond disorder. Markov Process. Related Fields 13 (2007) 519-542. | MR 2357386 | Zbl 1144.60058

[2] E. A. Carlen, S. Kusuoka and D. W. Stroock. Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 245-287. | Numdam | MR 898496 | Zbl 0634.60066

[3] E. B. Davies. Explicit constants for Gaussian upper bounds on heat kernels. Amer. J. Math. 109 (1987) 319-333. | MR 882426 | Zbl 0659.35009

[4] A. Faggionato and F. Martinelli. Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields 127 (2003) 535-608. | MR 2021195 | Zbl 1052.60083

[5] P. A. Ferrari, E. Presutti, E. Scacciatelli and M. E. Vares. The symmetric simple exclusion process I: Probability estimates. Stochastic Process. Appl. 39 (1991) 89-105. | MR 1135087 | Zbl 0749.60094

[6] G. Gielis, A. Koukkous and C. Landim. Equilibrium fluctuations for zero range processes in random environment. Stochastic Process. Appl. 77 (1998) 187-205. | MR 1649004 | Zbl 0935.60082

[7] R. A. Holley and D. W. Stroock. Generalized Onstein-Uhlenbeck processes and infinite branching Brownian motions. Kyoto Univ. R.I.M.S 14 (1978) 741-814. | MR 527199 | Zbl 0412.60065

[8] M. D. Jara and C. Landim. Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 567-577. | Numdam | MR 2259975 | Zbl 1101.60080

[9] C. Kipnis and C. Landim. Scaling Limit of Interacting Particles. Springer, Berlin, 1999. | MR 1707314

[10] C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Comm. Math. Phys. 106 (1986) 1-19. | MR 834478 | Zbl 0588.60058

[11] R. Kunnemann. The diffusion limit for reversible jump processes on ℤd with ergodic random bond conductivities. Comm. Math. Phys. 90 (1983) 27-68. | MR 714611 | Zbl 0523.60097

[12] C. Landim, S. Olla and S. R. S. Varadhan. Asymptotic behavior of a tagged particle in simple exclusion processes. Bol. Soc. Bras. Mat. 31 (2000) 241-275. | MR 1817088 | Zbl 0983.60100

[13] C. Landim and S. Volchan. Equilibrium fluctuations for driven tracer particle dynamics. Stochastic Process. Appl. 85 (2000) 139-158. | MR 1730614 | Zbl 0997.60121

[14] T. M. Liggett. Interacting Particle Systems. Springer, New York, 1985. | MR 776231 | Zbl 0559.60078

[15] I. Mitoma. Tightness of probabilities on C([0, 1]; S') and D([0, 1]; S'). Ann. Probab. 11 (1983) 989-999. | MR 714961 | Zbl 0527.60004

[16] K. Nagy. Symmetric random walk in random environment in one dimension. Period. Math. Hungar. 45 (2002) 101-120. | MR 1955197 | Zbl 1064.60202

[17] H. Rost and M. E. Vares. Hydrodynamics of a one-dimensional nearest neighbor model. Contemp. Math. 41 (1985) 329-342. | MR 814722 | Zbl 0572.60095

[18] V. Sidoravicius and A. Sznitman. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2004) 219-244. | MR 2063376 | Zbl 1070.60090

[19] H. Spohn. Large Scale Dynamics of Interacting Particles. Springer, Berlin, 1991. | Zbl 0742.76002

[20] A. Sznitman. Random motions in random media. In Mathematical Statistical Physics 219-242. A. Bovier, F. Dunlop, F. den Hollander, A. van Enter and J. Dalibard (Eds). Les Houches, Session LXXXIII, 2005, Elsevier, 2005.

[21] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Springer, Berlin-New York, 1979. | MR 532498 | Zbl 0426.60069