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Regularity of diffusion coefficient for nearest neighbor asymmetric simple exclusion on \(\mathbb{Z}\). (English) Zbl 1073.60091

Summary: We consider the nearest neighbor asymmetric exclusion process on \(\mathbb{Z}\), in which particles jump with probability \(p(1)\) to the right and \(p(-1)\) to the left. Let \(q=p(1)/p(-1)\) and denote by \(v_q\) an ergodic component of the reversible Bernoulli product measure which places a particle at \(x\) with probability \(q^x/(1+q^x)\). It is well known that under some hypotheses on a local function \(V\), \((1/\sqrt t)\int^t_0 V(\eta_s)ds\) converges to a normal distribution with variance \(\sigma^2 =\sigma^2(q)\), which depends on \(q\). We prove that \(\sigma^2(q)\) is a \(C^\infty\) function of \(q\) on \((0,1)\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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[1] Benjamini, I.; Ferrari, P. A.; Landim, C., Asymmetric conservative processes with random rates, Stochastic Process. Appl., 61, 2, 181-204 (1996) · Zbl 0849.60093
[2] Bernardin, C., Regularity of the diffusion coefficient for lattice gas reversible under Bernoulli measures, Stochastic Process. Appl., 101, 1, 43-68 (2002) · Zbl 1075.60580
[3] Caputo, P.; Martinelli, F., Asymmetric diffusion and the energy gap above 111 ground state of the quantum XXZ model, Comm. Math. Phys., 226, 2, 323-375 (2002) · Zbl 0990.82002
[4] Eyink Gregory, L.; Lebowitz Joel, L.; Herbert, S., Hydrodynamics and fluctuations outside of local equilibrium: driven diffusive systems, J. Statist. Phys., 83, 3-4, 385-472 (1996) · Zbl 1081.82595
[5] Kipnis, C.; Varadhan, S. R.S., Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion, Comm. Math. Phys., 104, 1, 1-19 (1986) · Zbl 0588.60058
[6] Landim, C.; Olla, S.; Varadhan, S. R.S., Asymptotic behavior of a tagged particle in simple exclusion processes, Bol. Soc. Brasil. Mat., 31, 3, 241-275 (2000) · Zbl 0983.60100
[7] Landim, C.; Olla, S.; Varadhan, S. R.S., Symmetric simple exclusion process: regularity of the self-diffusion coefficient, Comm. Math. Phys., 224, 1, 307-321 (2001) · Zbl 0994.60093
[8] Landim, C.; Olla, S.; Varadhan, S. R.S., On viscosity and fluctuation-dissipation in exclusion process, J. Statist. Phys., 115, 1-2, 323-363 (2004) · Zbl 1157.82355
[9] Liggett, T. M., Interacting Particle Systems (1985), Springer: Springer Berlin, New York · Zbl 0559.60078
[10] Y. Nagahata, Regularity of the diffusion coefficient matrix for the lattice gas with energy in dimensions \(d \geqs; 3\); Y. Nagahata, Regularity of the diffusion coefficient matrix for the lattice gas with energy in dimensions \(d \geqs; 3\) · Zbl 1129.60096
[11] Y. Nagahata, Regularity of the diffusion coefficient matrix for the lattice gas with energy in dimensions \(d = 1, 2\); Y. Nagahata, Regularity of the diffusion coefficient matrix for the lattice gas with energy in dimensions \(d = 1, 2\) · Zbl 1129.60096
[12] R.M. Sued, Regularity properties of the diffusion coefficient for a mean zero exclusion process, preprint.; R.M. Sued, Regularity properties of the diffusion coefficient for a mean zero exclusion process, preprint. · Zbl 1073.60098
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