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Asymmetric simple exclusion process with open boundaries and quadratic harnesses. (English) Zbl 1376.82055
Summary: We show that the joint probability generating function of the stationary measure of a finite state asymmetric exclusion process with open boundaries can be expressed in terms of joint moments of Markov processes called quadratic harnesses. We use our representation to prove the large deviations principle for the total number of particles in the system. We use the generator of the Markov process to show how explicit formulas for the average occupancy of a site arise for special choices of parameters. We also give similar representations for limits of stationary measures as the number of sites tends to infinity.

MSC:
82C22 Interacting particle systems in time-dependent statistical mechanics
60F10 Large deviations
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
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[1] Anshelevich, M, Free martingale polynomials, J. Funct. Anal., 201, 228-261, (2003) · Zbl 1033.46050
[2] Bertini, L; Sole, A; Gabrielli, D; Jona-Lasinio, G; Landim, C, Macroscopic fluctuation theory for stationary non-equilibrium states, J. Stat. Phys., 107, 635-675, (2002) · Zbl 1031.82038
[3] Biane, P, Quelques proprietes du mouvement brownien non-commutatif, Astérisque, 236, 73-102, (1996) · Zbl 0867.46043
[4] Biane, P, Processes with free increments, Math. Z., 227, 143-174, (1998) · Zbl 0902.60060
[5] Blythe, R; Evans, M; Colaiori, F; Essler, F, Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra, J. Phys. A, 33, 2313-2332, (2000) · Zbl 1100.82512
[6] Blythe, RA; Evans, MR, Nonequilibrium steady states of matrix-product form: a solver’s guide, J. Phys. A, 40, r333-r441, (2007) · Zbl 1155.82325
[7] Bożejko, M; Kümmerer, B; Speicher, R, \(q\)-Gaussian processes: non-commutative and classical aspects, Commun. Math. Phys., 185, 129-154, (1997) · Zbl 0873.60087
[8] Bryc, W; Matysiak, W; Wesołowski, J, Quadratic harnesses, \(q\)-commutations, and orthogonal martingale polynomials, Trans. Am. Math. Soc., 359, 5449-5483, (2007) · Zbl 1129.60068
[9] Bryc, W; Matysiak, W; Wesołowski, J, The bi-Poisson process: a quadratic harness, Ann. Probab., 36, 623-646, (2008) · Zbl 1137.60036
[10] Bryc, W; Wesołowski, J, Classical bi-Poisson process: an invertible quadratic harness, Stat. Probab. Lett., 76, 1664-1674, (2006) · Zbl 1105.60053
[11] Bryc, W; Wesołowski, J, Bi-Poisson process, Infinite Dimens. Anal. Quantum Probab. Relat. Top., 10, 277-291, (2007) · Zbl 1118.60066
[12] Bryc, W; Wesołowski, J, Askey-Wilson polynomials, quadratic harnesses and martingales, Ann. Probab., 38, 1221-1262, (2010) · Zbl 1201.60077
[13] Bryc, W; Wesołowski, J, Infinitesimal generators of \(q\)-meixner processes, Stoch. Proces. Appl., 124, 915-926, (2014) · Zbl 1300.60091
[14] Bryc, W; Wesołowski, J, Infinitesimal generators for a class of polynomial processes, Stud. Math., 229, 73-93, (2015) · Zbl 1343.60119
[15] Corteel, S; Stanley, R; Stanton, D; Williams, L, Formulae for Askey-Wilson moments and enumeration of staircase tableaux, Trans. Am. Math. Soc., 364, 6009-6037, (2012) · Zbl 1269.05116
[16] Corteel, S; Williams, LK, A Markov chain on permutations which projects to the PASEP, Int. Math. Res. Notices, 2007, rnm055, (2077) · Zbl 1132.60070
[17] Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, vol. 38. Springer, New York (2009) · Zbl 0896.60013
[18] Derrida, B; Domany, E; Mukamel, D, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys., 69, 667-687, (1992) · Zbl 0893.60077
[19] Derrida, B; Douçot, B; Roche, P-E, Current fluctuations in the one-dimensional symmetric exclusion process with open boundaries, J. Stat. Phys., 115, 717-748, (2004) · Zbl 1052.82020
[20] Derrida, B; Evans, M; Hakim, V; Pasquier, V, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A, 26, 1493-1517, (1993) · Zbl 0772.60096
[21] Derrida, B; Lebowitz, J; Speer, E, Free energy functional for nonequilibrium systems: an exactly solvable case, Phys. Rev. Lett., 87, 150601, (2001)
[22] Derrida, B; Lebowitz, J; Speer, E, Exact free energy functional for a driven diffusive open stationary nonequilibrium system, Phys. Rev. Lett., 89, 030601, (2002) · Zbl 1031.60083
[23] Derrida, B; Lebowitz, J; Speer, E, Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process, J. Stat. Phys., 110, 775-810, (2003) · Zbl 1031.60083
[24] Duhart, H. G., Mörters, P., Zimmer, J.: The semi-infinite asymmetric exclusion process: Large deviations via matrix products (2014). arXiv:1411.3270 · Zbl 1052.82020
[25] Enaud, C; Derrida, B, Large deviation functional of the weakly asymmetric exclusion process, J. Stat. Phys., 114, 537-562, (2004) · Zbl 1061.82020
[26] Essler, FH; Rittenberg, V, Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries, J. Phys. A, 29, 3375-3407, (1996) · Zbl 0902.60088
[27] González Duhart Muñoz de Cote, H.: Large Deviations for Boundary Driven Exclusion Processes. PhD thesis, University of Bath (2015) · Zbl 1327.60144
[28] Großkinsky, S.: Phase transitions in nonequilibrium stochastic particle systems with local conservation laws. PhD thesis, Technical University of Munich (2004) · Zbl 1132.60070
[29] Johnston, D., Stringer, M.: The PASEP at \(q=-1\). (2012). arXiv:1207.7316 · Zbl 1047.82019
[30] Josuat-Vergès, M.: Combinatorics of the three-parameter PASEP partition function. Electron. J. Combin. 18(1), #P22 (2011a) · Zbl 1205.05011
[31] Josuat-Vergès, M, Rook placements in Young diagrams and permutation enumeration, Adv. Appl. Math., 47, 1-22, (2011) · Zbl 1225.05022
[32] Kim, JS; Stanton, D, Moments of Askey-Wilson polynomials, J. Combin. Theory Ser. A, 125, 113-145, (2014) · Zbl 1295.05034
[33] Liggett, TM, Ergodic theorems for the asymmetric simple exclusion process, Trans. Am. Math. Soc., 213, 237-261, (1975) · Zbl 0322.60086
[34] Matysiak, W; Świeca, M, Zonal polynomials and a multidimensional quantum Bessel process, Stoch. Process. Appl., 125, 3430-3457, (2015) · Zbl 1327.60144
[35] Sandow, S, Partially asymmetric exclusion process with open boundaries, Phys. Rev. E, 50, 2660-2667, (1994)
[36] Sasamoto, T, One-dimensional partially asymmetric simple exclusion process with open boundaries: orthogonal polynomials approach, J. Phys. A, 32, 7109, (1999) · Zbl 0962.82020
[37] Sasamoto, T., Williams, L.: Combinatorics of the asymmetric exclusion process on a semi-infinite lattice (2012). arXiv preprint arXiv:1204.1114 · Zbl 1325.60163
[38] Schoutens, W.: Stochastic processes and orthogonal polynomials. Springer Verlag, New York (2000) · Zbl 0960.60076
[39] Schütz, G; Domany, E, Phase transitions in an exactly soluble one-dimensional exclusion process, J. Stat. Phys., 72, 277-296, (1993) · Zbl 1099.82506
[40] Spitzer, F, Interaction of Markov processes, Adv. Math., 5, 246-290, (1970) · Zbl 0312.60060
[41] Szabłowski, PJ, Moments of \(q\)-normal and conditional \(q\)-normal distributions, Stat. Probab. Lett., 106, 65-72, (2015) · Zbl 1463.62020
[42] Uchiyama, M; Sasamoto, T; Wadati, M, Asymmetric simple exclusion process with open boundaries and Askey-Wilson polynomials, J. Phys. A, 37, 4985-5002, (2004) · Zbl 1047.82019
[43] Uchiyama, M; Wadati, M, Correlation function of asymmetric simple exclusion process with open boundaries, J. Nonlinear Math. Phys., 12, 676-688, (2005) · Zbl 1362.82028
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