Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries

Włodzimierz Bryc and Yizao Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We investigate the fluctuations of cumulative density of particles in the asymmetric simple exclusion process with respect to the stationary distribution (also known as the steady state), as a stochastic process indexed by $[0,1]$. In three phases of the model and their boundaries within the fan region, we establish a complete picture of the scaling limits of the fluctuations of the density as the number of sites goes to infinity. In the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian excursion. This extends an earlier result by Derrida et al. (J. Statist. Phys. 115 (2004) 365–382) for totally asymmetric simple exclusion process in the same phase. In the low/high density phases, the limit fluctuations are Brownian motion. Most interestingly, at the boundary of the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian meander (or a time-reversal of the latter, depending on the side of the boundary). Our proofs rely on a representation of the joint generating function of the asymmetric simple exclusion process with respect to the stationary distribution in terms of joint moments of a Markov processes, which is constructed from orthogonality measures of the Askey–Wilson polynomials.

Résumé

Nous étudions les fluctuations de la densité de particules dans un processus d’exclusion simple asymétrique sous la distribution stationnaire (ou état stable), vues comme un processus stochastique indexé par $[0,1]$. Pour trois des phases du modèle et à leurs frontières nous obtenons une description complète des limites d’échelles de ces fluctuations lorsque le nombre de sites tend vers l’infini. Dans la phase de courant maximal, la limite est la somme de deux processus indépendants : un mouvement brownien et une excursion brownienne. Ce résultat étend celui obtenu précédemment par Derrida et al. (J. Statist. Phys. 115 (2004) 365–382) pour le processus d’exclusion simple totalement asymétrique et dans la même phase. Dans les phases de fortes et faibles densités, les limites sont des mouvements browniens. De façon plus intéressante, à la frontière de la phase de courant maximal, la limite est la somme de deux processus indépendants : un mouvement brownien et un méandre brownien (ou, selon la partie de la frontière, un méandre brownien renversé en temps). Nos démonstrations reposent sur une représentation des fonctions génératrices des lois fini-dimensionnelles du processus d’exclusion simple asymétrique en termes de moments joints d’un processus de Markov construit à partir de mesures rendant orthogonaux les polynômes d’Askey–Wilson.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 4 (2019), 2169-2194.

Dates
Received: 30 April 2018
Revised: 11 October 2018
Accepted: 18 October 2018
First available in Project Euclid: 8 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1573203626

Digital Object Identifier
doi:10.1214/18-AIHP945

Mathematical Reviews number (MathSciNet)
MR4029151

Subjects
Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Asymmetric simple exclusion process Scaling limit Phase transition Askey–Wilson process Brownian excursion Brownian meander Laplace transform Tangent process

Citation

Bryc, Włodzimierz; Wang, Yizao. Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 4, 2169--2194. doi:10.1214/18-AIHP945. https://projecteuclid.org/euclid.aihp/1573203626


Export citation

References

  • [1] R. Askey and J. Wilson. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (1985), iv+55.
  • [2] P. Biane. Processes with free increments. Math. Z. 227 (1998) 143–174.
  • [3] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, 1999.
  • [4] R. A. Blythe and M. R. Evans. Nonequilibrium steady states of matrix-product form: A solver’s guide. J. Phys. A 40 (2007) R333–R441.
  • [5] R. A. Blythe, M. R. Evans, F. Colaiori and F. H. L. Essler. Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra. J. Phys. A 33 (2000) 2313–2332.
  • [6] W. Bryc, W. Matysiak and J. Wesołowski. Quadratic harnesses, $q$-commutations, and orthogonal martingale polynomials. Trans. Amer. Math. Soc. 359 (2007) 5449–5483.
  • [7] W. Bryc and Y. Wang. The local structure of $q$-Gaussian processes. Probab. Math. Statist. 36 (2016) 335–352.
  • [8] W. Bryc and Y. Wang. Dual representations of Laplace transforms of Brownian excursion and generalized meanders. Statist. Probab. Lett. 140 (2018) 77–83.
  • [9] W. Bryc and J. Wesołowski. Askey–Wilson polynomials, quadratic harnesses and martingales. Ann. Probab. 38 (2010) 1221–1262.
  • [10] W. Bryc and J. Wesołowski. Asymmetric simple exclusion process with open boundaries and quadratic harnesses. J. Stat. Phys. 167 (2017) 383–415.
  • [11] S. Corteel, M. Josuat-Vergès and L. K. Williams. The matrix ansatz, orthogonal polynomials, and permutations. Adv. in Appl. Math. 46 (2011) 209–225.
  • [12] I. Corwin and H. Shen. Open ASEP in the weakly asymmetric regime. Comm. Math. Phys. 71 (4) (2018) 2065–2128.
  • [13] J. H. Curtiss. A note on the theory of moment generating functions. Ann. Math. Stat. 13 (1942) 430–433.
  • [14] J. de Gier and F. H. Essler. Large deviation function for the current in the open asymmetric simple exclusion process. Phys. Rev. Lett. 107 (2011), Article ID 010602.
  • [15] B. Derrida. Matrix ansatz large deviations of the density in exclusion processes. In International Congress of Mathematicians. Vol. III 367–382. Eur. Math. Soc., Zürich, 2006.
  • [16] B. Derrida. Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current. J. Stat. Mech. Theory Exp. 2007 (2007), Article ID P07023.
  • [17] B. Derrida, E. Domany and D. Mukamel. An exact solution of a one-dimensional asymmetric exclusion model with open boundaries. J. Stat. Phys. 69 (1992) 667–687.
  • [18] B. Derrida, C. Enaud, C. Landim and S. Olla. Fluctuations in the weakly asymmetric exclusion process with open boundary conditions. J. Stat. Phys. 118 (2005) 795–811.
  • [19] B. Derrida, C. Enaud and J. L. Lebowitz. The asymmetric exclusion process and Brownian excursions. J. Stat. Phys. 115 (2004) 365–382.
  • [20] B. Derrida, M. R. Evans, V. Hakim and V. Pasquier. Exact solution of a $1$D asymmetric exclusion model using a matrix formulation. J. Phys. A 26 (1993) 1493–1517.
  • [21] B. Derrida, J. L. Lebowitz and E. R. Speer. Exact free energy functional for a driven diffusive open stationary nonequilibrium system. Phys. Rev. Lett. 89 (2002), Article ID 030601.
  • [22] B. Derrida, J. L. Lebowitz and E. R. Speer. Exact large deviation functional of a stationary open driven diffusive system: The asymmetric exclusion process. J. Stat. Phys. 110 (2003) 775–810. Special issue in honor of Michael E. Fisher’s 70th birthday (Piscataway, NJ, 2001).
  • [23] R. T. Durrett, D. L. Iglehart and D. R. Miller. Weak convergence to Brownian meander and Brownian excursion. Ann. Probab. 5 (1977) 117–129.
  • [24] C. Enaud and B. Derrida. Large deviation functional of the weakly asymmetric exclusion process. J. Stat. Phys. 114 (2004) 537–562.
  • [25] F. H. L. Essler and V. Rittenberg. Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries. J. Phys. A 29 (1996) 3375–3407.
  • [26] G. Eyink, J. L. Lebowitz and H. Spohn. Hydrodynamics of stationary nonequilibrium states for some stochastic lattice gas models. Comm. Math. Phys. 132 (1990) 253–283.
  • [27] G. Eyink, J. L. Lebowitz and H. Spohn. Lattice gas models in contact with stochastic reservoirs: Local equilibrium and relaxation to the steady state. Comm. Math. Phys. 140 (1991) 119–131.
  • [28] K. J. Falconer. The local structure of random processes. J. Lond. Math. Soc. (2) 67 (2003) 657–672.
  • [29] R. H. Farrell. Techniques of Multivariate Calculation. Lecture Notes in Mathematics 520. Springer, Berlin–New York, 1976.
  • [30] P. Gonçalves, C. Landim and A. Milanés. Nonequilibrium fluctuations of one-dimensional boundary driven weakly asymmetric exclusion processes. Ann. Appl. Probab. 27 (2017) 140–177.
  • [31] J. Hoffmann-Jørgensen. Probability with a View Toward Statistics, Vol. 1. Chapman & Hall, New York, 1994.
  • [32] S. Janson. Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas. Probab. Surv. 4 (2007) 80–145.
  • [33] R. Koekoek and R. F. Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue. Report no 98-17, Department of Technical Mathematics and Informatics, Faculty of Information Technology and Systems, Delft University of Technology, 1998.
  • [34] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer-Verlag, New York, 1985.
  • [35] C. T. MacDonald, J. H. Gibbs and A. C. Pipkin. Kinetics of biopolymerization on nucleic acid templates. Biopolymers 6 (1968) 1–25.
  • [36] A. Mukherjea, M. Rao and S. Suen. A note on moment generating functions. Statist. Probab. Lett. 76 (2006) 1185–1189.
  • [37] J. Pitman. Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electron. J. Probab. 4 (11) (1999) Article ID 33.
  • [38] J. Pitman. Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875. Springer-Verlag, Berlin, 2006.
  • [39] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer-Verlag, Berlin, 1999.
  • [40] S. Sandow. Partially asymmetric exclusion process with open boundaries. Phys. Rev. E 50 (1994) 2660–2667.
  • [41] T. Sasamoto. One-dimensional partially asymmetric simple exclusion process with open boundaries: Orthogonal polynomials approach. J. Phys. A 32 (1999) 7109–7131.
  • [42] G. Schütz and E. Domany. Phase transitions in an exactly soluble one-dimensional exclusion process. J. Stat. Phys. 72 (1993) 277–296.
  • [43] F. Spitzer. Interaction of Markov processes. Adv. Math. 5 (1970) 246–290.
  • [44] M. Uchiyama, T. Sasamoto and M. Wadati. Asymmetric simple exclusion process with open boundaries and Askey–Wilson polynomials. J. Phys. A 37 (2004) 4985–5002.
  • [45] M. Uchiyama and M. Wadati. Correlation function of asymmetric simple exclusion process with open boundaries. J. Nonlinear Math. Phys. 12 (2005) 676–688.
  • [46] Y. Wang. Extremes of $q$-Ornstein–Uhlenbeck processes. Stochastic Process. Appl. 128 (2018) 2979–3005.
  • [47] J.-Y. Yen and M. Yor. Local Times and Excursion Theory for Brownian Motion. Lecture Notes in Mathematics 2088. Springer, Cham, 2013.