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Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries. (English. French summary) Zbl 1434.60066
Summary: 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 B. Derrida et al. [J. Stat. Phys. 115, No. 1–2, 365–382 (2004; Zbl 1157.82354)] 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.

MSC:
60F05 Central limit and other weak theorems
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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References:
[1] R. Askey and J. Wilson. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (1985), iv+55. · Zbl 0572.33012
[2] P. Biane. Processes with free increments. Math. Z. 227 (1998) 143-174. · Zbl 0902.60060
[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. · Zbl 0944.60003
[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. · Zbl 1155.82325
[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. · Zbl 1100.82512
[6] W. Bryc, W. Matysiak and J. Wesołowski. Quadratic harnesses, \(q\)-commutations, and orthogonal martingale polynomials. Trans. Amer. Math. Soc. 359 (2007) 5449-5483. · Zbl 1129.60068
[7] W. Bryc and Y. Wang. The local structure of \(q\)-Gaussian processes. Probab. Math. Statist. 36 (2016) 335-352. · Zbl 1357.60081
[8] W. Bryc and Y. Wang. Dual representations of Laplace transforms of Brownian excursion and generalized meanders. Statist. Probab. Lett. 140 (2018) 77-83. · Zbl 1391.60180
[9] W. Bryc and J. Wesołowski. Askey-Wilson polynomials, quadratic harnesses and martingales. Ann. Probab. 38 (2010) 1221-1262. · Zbl 1201.60077
[10] W. Bryc and J. Wesołowski. Asymmetric simple exclusion process with open boundaries and quadratic harnesses. J. Stat. Phys. 167 (2017) 383-415. · Zbl 1376.82055
[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. · Zbl 1227.05036
[12] I. Corwin and H. Shen. Open ASEP in the weakly asymmetric regime. Comm. Math. Phys. 71 (4) (2018) 2065-2128. · Zbl 1410.82014
[13] J. H. Curtiss. A note on the theory of moment generating functions. Ann. Math. Stat. 13 (1942) 430-433. · Zbl 0063.01024
[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. · Zbl 1099.60069
[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. · Zbl 1456.82551
[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. · Zbl 0893.60077
[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. · Zbl 1073.82029
[19] B. Derrida, C. Enaud and J. L. Lebowitz. The asymmetric exclusion process and Brownian excursions. J. Stat. Phys. 115 (2004) 365-382. · Zbl 1157.82354
[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. · Zbl 0772.60096
[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). · Zbl 1031.60083
[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. · Zbl 0356.60034
[24] C. Enaud and B. Derrida. Large deviation functional of the weakly asymmetric exclusion process. J. Stat. Phys. 114 (2004) 537-562. · Zbl 1061.82020
[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. · Zbl 0902.60088
[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. · Zbl 0706.76082
[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. · Zbl 0734.60110
[28] K. J. Falconer. The local structure of random processes. J. Lond. Math. Soc. (2) 67 (2003) 657-672. · Zbl 1054.28003
[29] R. H. Farrell. Techniques of Multivariate Calculation. Lecture Notes in Mathematics 520. Springer, Berlin-New York, 1976. · Zbl 0337.62033
[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. · Zbl 1362.60085
[31] J. Hoffmann-Jørgensen. Probability with a View Toward Statistics, Vol. 1. Chapman & Hall, New York, 1994. · Zbl 0821.62003
[32] S. Janson. Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas. Probab. Surv. 4 (2007) 80-145. · Zbl 1189.60147
[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. · Zbl 0559.60078
[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. · Zbl 1092.60008
[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. · Zbl 0935.60068
[38] J. Pitman. Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875. Springer-Verlag, Berlin, 2006. · Zbl 1103.60004
[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. · Zbl 0917.60006
[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. · Zbl 0962.82020
[42] G. Schütz and E. Domany. Phase transitions in an exactly soluble one-dimensional exclusion process. J. Stat. Phys. 72 (1993) 277-296. · Zbl 1099.82506
[43] F. Spitzer. Interaction of Markov processes. Adv. Math. 5 (1970) 246-290. · Zbl 0312.60060
[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. · Zbl 1047.82019
[45] M. Uchiyama and M. Wadati. Correlation function of asymmetric simple exclusion process with open boundaries. J. Nonlinear Math. Phys. 12 (2005) 676-688. · Zbl 1362.82028
[46] Y. Wang. Extremes of \(q\)-Ornstein-Uhlenbeck processes. Stochastic Process. Appl. 128 (2018) 2979-3005. · Zbl 1405.60072
[47] J.-Y. Yen and M. Yor. Local Times and Excursion Theory for Brownian Motion. Lecture Notes in Mathematics 2088. Springer, Cham, 2013. · Zbl 1364.60003
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