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Correlation inequalities for interacting particle systems with duality. (English) Zbl 1202.82040

Summary: We prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other – a process which we call here the symmetric inclusion process (SIP) – or repel each other – a generalized version of the well-known symmetric exclusion process. As an application, new correlation inequalities are obtained for the SIP, as well as for some interacting diffusions which are used as models of heat conduction – the so-called Brownian momentum process, and the Brownian energy process. These inequalities are counterparts of the inequalities (in the opposite direction) for the symmetric exclusion process, showing that the SIP is a natural bosonic analogue of the symmetric exclusion process, which is fermionic. Finally, we consider a boundary driven version of the SIP for which we prove duality and then obtain correlation inequalities.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
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[1] Andjel, E.: A correlation inequality for the symmetric exclusion process. Ann. Probab. 16, 717–721 (1988) · Zbl 0642.60105 · doi:10.1214/aop/1176991782
[2] Bernardin, C.: Superdiffusivity of asymmetric energy model in dimensions 1 and 2. J. Math. Phys. 49(10), 103301 (2008) · Zbl 1152.81335 · doi:10.1063/1.3000580
[3] Bernardin, C., Olla, S.: Fourier’s law for a microscopic model of heat conduction. J. Stat. Phys. 121, 271–289 (2005) · Zbl 1127.82042 · doi:10.1007/s10955-005-7578-9
[4] Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Stochastic interacting particle systems out of equilibrium. J. Stat. Mech., Theory Exp. P07014n (2007) · Zbl 1190.82027
[5] Borcea, J., Brändén, P., Liggett, T.M.: Negative dependence and the geometry of polynomials. J. Am. Math. Soc. 22, 521–567 (2009) · Zbl 1206.62096 · doi:10.1090/S0894-0347-08-00618-8
[6] De Masi, A., Presutti, E.: Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Mathematics, vol. 1501. Springer, Berlin (1991) · Zbl 0754.60122
[7] Derrida, B., Lebowitz, J.L., Speer, E.R.: Entropy of open lattice systems. J. Stat. Phys. 126, 1083–1108 (2007) · Zbl 1153.82013 · doi:10.1007/s10955-006-9160-5
[8] Galves, A., Kipnis, C., Marchioro, C., Presutti, E.: Nonequilibrium measures which exhibit a temperature gradient: study of a model. Commun. Math. Phys. 81, 127–147 (1981) · Zbl 0465.60089 · doi:10.1007/BF01941803
[9] Giardina, C., Kurchan, J.: The Fourier law in a momentum-conserving chain. J. Stat. Mech. P05009 (2005)
[10] Giardina, C., Kurchan, J., Redig, F.: Duality and exact correlations for a model of heat conduction. J. Math. Phys. 48, 033301 (2007) · Zbl 1137.82332 · doi:10.1063/1.2711373
[11] Giardina, C., Kurchan, J., Redig, F., Vafayi, K.: Duality and hidden symmetries in interacting particle systems. J. Stat. Phys. 135, 25–55 (2009) · Zbl 1173.82020 · doi:10.1007/s10955-009-9716-2
[12] Gobron, T., Saada, E.: Coupling, attractiveness and hydrodynamics for conservative particle systems. Preprint available on arxiv.org (2009) · Zbl 1252.60093
[13] Harris, T.E.: A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probab. 5, 451–454 (1977) · Zbl 0381.60072 · doi:10.1214/aop/1176995804
[14] Inglis, J., Neklyudov, M., Zegarlinski, B.: Ergodicity for infinite particle systems with locally conserved quantities. arXiv:1002.0282v2 (2010) · Zbl 1266.60157
[15] Liggett, T.M.: Negative correlations and particle systems. Markov Process. Relat. Fields 8, 547–564 (2002) · Zbl 1021.60084
[16] Liggett, T.M.: Interacting Particle Systems. Classics in Mathematics. Springer, Berlin (2005). Reprint of the 1985 original · Zbl 0559.60078
[17] Spohn, H.: Long range correlations for stochastic lattice gases in a non-equilibrium steady state. J. Phys. A 16, 4275–4291 (1983) · doi:10.1088/0305-4470/16/18/029
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