Journal of Statistical Physics

, Volume 171, Issue 5, pp 802–821 | Cite as

The Central Limit Theorem for Supercritical Oriented Percolation in Two Dimensions

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Abstract

We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the standard normal law. This resolves a longstanding open problem pointed out to in several instances in the literature. The result applies also to the continuous-time analog of the process, viz. the basic one-dimensional contact process. We also derive general random-indices central limit theorems for associated random variables as byproducts of our proof.

Keywords

Oriented bond percolation Central limit theorems Association Contact process 

Mathematics Subject Classification

Primary 60K35 Secondary 82B43 

Notes

Acknowledgements

This work has been supported during non-overlapping periods of time by CONICET, by FAPESP grant 2016/03988-5, and, currently, by PNPD/CAPES.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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