Abstract
We prove that, once an algorithm of perfect simulation for a stationary and ergodic random field F taking values in \(S^{\mathbb{Z}^{d}}\), S a bounded subset of R n, is provided, the speed of convergence in the mean ergodic theorem occurs exponentially fast for F. Applications from (non-equilibrium) statistical mechanics and interacting particle systems are presented.
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Baum, L.E., Katz, M., Read, R.R.: Exponential convergence rates for the law of large numbers. Trans. Am. Math. Soc. 102, 187–199 (1962)
Ferrari, P.A., Fernandez, R., Garcia, N.L.: Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Appl. 102, 63–88 (2002)
Galves, A., Garcia, N.L., Löcherbach, E.: Perfect simulation and finitary coding for multicolor systems with interactions of infinite range. Preprint
Häggström, O., Steif, J.E.: Propp-Wilson algorithms and finitary codings for high noise Markov random fields. Comb. Probab. Comput. 9, 425–439 (2000)
Lebowitz, J.L., Schonmann, R.H.: Pseudo-free energies and large deviations for non Gibbsian FKG measures. Probab. Theory Relat. Fields 77, 49–64 (1988)
Olla, S.: Large deviations for Gibbs random fields. Probab. Theory Relat. Fields 77, 343–357 (1988)
Propp, J.G., Wilson, D.B.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Algorithms 9, 223–252 (1996)
Ritchie, T.L.: Construction of the thermodynamic jamming limit for the parking process and other exclusion schemes on ℤd. J. Stat. Phys. 122(3), 381–398 (2006)
van den Berg, J., Steif, J.E.: On the existence and non-existence of finitary codings for a class of random fields. Ann. Probab. 27, 1501–1522 (1999)
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G.G. Bosco is supported by a grant from Capes.
T.L. Ritchie is supported by grants from CNPq and Fapesp.
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Bosco, G.G., Machado, F.P. & Ritchie, T.L. Exponential Rates of Convergence in the Ergodic Theorem: A Constructive Approach. J Stat Phys 139, 367–374 (2010). https://doi.org/10.1007/s10955-010-9945-4
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DOI: https://doi.org/10.1007/s10955-010-9945-4