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The semi-infinite asymmetric exclusion process: large deviations via matrix products. (English) Zbl 1459.60068
Summary: We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. T. M. Liggett [Trans. Am. Math. Soc. 213, 237–261 (1975; Zbl 0322.60086)] has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source and the original density are below a critical value, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of B. Derrida et al. [J. Phys. A, Math. Gen. 26, No. 7, 1493–1517 (1993; Zbl 0772.60096)] it was shown by S. Grosskinsky [Phase transitions in nonequilibrium stochastic particle systems with local conservation laws. Munich: TU Munich (PhD Thesis) (2004)] that these correlations can be described by means of a matrix product representation. In this paper we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and is potentially applicable to other models described by matrix products.

MSC:
60F10 Large deviations
37K06 General theory of infinite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, conservation laws
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
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[1] Angeletti, F., Touchette, H., Bertin, E., Abry, P.: Large deviations for correlated random variables described by a matrix product ansatz. J. Stat. Mech. Theory Exp. 2014(2), P02003, 17 (2014) · Zbl 0322.60086
[2] Bahadoran, C.: A quasi-potential for conservation laws with boundary conditions. arXiv:1010.3624 (2010) · Zbl 0901.60086
[3] Bertini, L; De Sole, A; Gabrielli, D; Jona-Lasinio, G; Landim, C, Large deviations for the boundary driven symmetric simple exclusion process, Math. Phys. Anal. Geom., 6, 231-267, (2003) · Zbl 1031.82039
[4] Bertini, L; De Sole, A; Gabrielli, D; Jona-Lasinio, G; Landim, C, Large deviation approach to non equilibrium processes in stochastic lattice gases, Bull. Braz. Math. Soc., 37, 611-643, (2006) · Zbl 1109.60325
[5] Bertini, L; Landim, C; Mourragui, M, Dynamical large deviations for the boundary driven weakly asymmetric exclusion process, Ann. Probab., 37, 2357-2403, (2009) · Zbl 1187.82083
[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] Bodineau, T; Giacomin, G, From dynamic to static large deviations in boundary driven exclusion particle systems, Stoch. Proc. Appl., 110, 67-81, (2004) · Zbl 1075.60122
[8] Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Volume 38 of Stochastic Modelling and Applied Probability. Springer, Berlin (2010) · Zbl 1177.60035
[9] den Hollander, F.: Large Deviations, Volume 14 of Fields Institute Monographs. American Mathematical Society, Providence (2000)
[10] Derrida, B.: Matrix ansatz large deviations of the density in exclusion processes. In: International Congress of Mathematicians, vol. III, pp. 367-382. Eur. Math. Soc., Zürich (2006) · Zbl 1099.60069
[11] Derrida, B., Evans, M. R., Hakim, V., Pasquier, V.: Exact solution of a 1d asymmetric exclusion model using a matrix formulation. J. Phys. A 26(7), 1493-1517 (1993) · Zbl 0772.60096
[12] Derrida, B; Lebowitz, JL; Speer, ER, Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process, J. Statist. Phys., 110, 775-810, (2003) · Zbl 1031.60083
[13] Evans, MR; Ferrari, PA; Mallick, K, Matrix representation of the stationary measure for the multispecies TASEP, J. Stat. Phys., 135, 217-239, (2009) · Zbl 1167.82017
[14] Grosskinsky, S.: Phase transitions in nonequilibrium stochastic particle systems with local conservation laws. PhD thesis, TU Munich (2004) · Zbl 1167.82017
[15] Hinrichsen, H, Matrix product ground states for exclusion processes with parallel dynamics, J. Phys. A, 29, 3659-3667, (1996) · Zbl 0901.60086
[16] Liggett, TM, Ergodic theorems for the asymmetric simple exclusion process, Trans. Am. Math. Soc., 213, 237-261, (1975) · Zbl 0322.60086
[17] Nyawo, PT; Touchette, H, A minimal model of dynamical phase transition, EPL (Europhys. Lett.), 116, 50009, (2016)
[18] Sasamoto, T, One-dimensional partially asymmetric simple exclusion process with open boundaries: orthogonal polynomials approach, J. Phys. A, 32, 7109-7131, (1999) · Zbl 0962.82020
[19] Sasamoto, T., Williams, L.: Combinatorics of the asymmetric exclusion process on a semi-infinite lattice. arXiv:1204.1114 (2012) · Zbl 1325.60163
[20] Stinchcombe, RB; Schütz, GM, Operator algebra for stochastic dynamics and the Heisenberg chain, EPL (Europhys. Lett.), 29, 663, (1995)
[21] Tracy, CA; Widom, H, The asymmetric simple exclusion process with an open boundary, J. Math. Phys., 54, 103301, 16, (2013) · Zbl 1288.82043
[22] Trefethen, L. N., Embree, M.: Spectra and Pseudospectra. Princeton University Press, Princeton (2005) · Zbl 1085.15009
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