Abstract
We prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles. It is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system. We show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density. This improves the result of Großkinsky, Schütz and Spohn, where convergence is obtained for the finite dimensional marginals. We obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk.
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Abramowitz M.: Handbook Mathematical Functions. Dover, New York (1972)
Baltrunas A.: On a local limit theorem on one-sided large deviations for dominated-variation distributions. Lithuanian Math. J. 36(1), 1–7 (1996)
Dembo A., Zeitouni O.: Refinements of the Gibbs conditioning principle. Prob. Th. Rel. Fields 104, 1–14 (1996)
Doney R.A.: A local limit theorem for moderate deviations. Bull. London Math. Soc. 33, 100–108 (2001)
Evans M.R.: Phase transitions in one-dimensional nonequilibrium systems. Braz. J. Phys. 30(1), 42–57 (2000)
Evans M.R., Hanney T.: Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A: Math. Gen. 38, 195–240 (2005)
Ferrari P., Landim C., Sisko V.: Condensation for a fixed number of independent random variables. J. Stat. Phys 128(5), 1153–1158 (2007)
Gnedenko B.V., Kolmogorov A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading (1949)
Godrèche C., Luck J.M.: Dynamics of the condensate in zero-range processes. J. Phys. A Math. Gen. 38, 7215–7237 (2005)
Großkinsky, S.: Equivalence of ensembles for two-component zero-range invariant measures to appear in Stoch. Proc. Appl., available from http://www.warwick.ac.uk/~masgav
Großkinsky S., Schütz G.M., Spohn H.: Condensation in the zero range process: stationary and dynamical properties. J. Stat. Phys. 113, 389–410 (2003)
Jeon I., March P., Pittel B.: Size of the largest cluster under zero-range invariant measures. Ann. Probab. 28, 1162–1194 (2000)
Kipnis C., Landim C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin Heidelburg (1999)
Liggett T.M.: An invariance principle for conditioned sums of independent random variables. J. Math. Mech. 18, 559–570 (1968)
Nagaev A.V.: Limit theorems that take into account large deviations when Cramér’s condition is violated (in Russian). Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 13(6), 17–22 (1969)
Nagaev A.V.: Local limit theorems with regard to large deviations when Cramér’s condition is not satisfied. Litovsk. Mat. Sb. 8, 553–579 (1968)
Skorokhod A.V.: Limit theorems for stochastic processes with independent increments. Theory Probab. Appl. 2, 138–171 (1957)
Spitzer F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970)
Tkačuk S.G.: Local limit theorems, allowing for large deviations, in the case of stable limit laws (in Russian). Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 17(2), 30–33 (1973)
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Armendáriz, I., Loulakis, M. Thermodynamic limit for the invariant measures in supercritical zero range processes. Probab. Theory Relat. Fields 145, 175–188 (2009). https://doi.org/10.1007/s00440-008-0165-7
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DOI: https://doi.org/10.1007/s00440-008-0165-7
Keywords
- Condensation
- Equivalence of ensembles
- Large deviations
- Subexponential distributions
- Zero range processes