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On Decoupling Inequalities and Percolation of Excursion Sets of the Gaussian Free Field

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Abstract

We prove decoupling inequalities for the Gaussian free field on \({\mathbb {Z}}^d\), \(d\ge 3\). As an application, we obtain exponential decay (with logarithmic correction for \(d=3\)) of the connectivity function of excursion sets for large values of the threshold.

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Acknowledgments

The work of Serguei Popov was partially supported by CNPq (300328/2005–2) and FAPESP (2009/52379–8). The authors also thank the organizers of the conference Random Walks: Crossroads and Perspectives (Budapest, June 24–28, 2013), for providing the opportunity for the authors to meet and work on this topic. This paper was written while B.R. was a postdoctoral fellow of the University of British Columbia. We thank Prof. Alain-Sol Sznitman for pointing out the reference [5] to us, moreover Pierre-François Rodriguez, Alexander Drewitz and a very thorough anonymous referee for reading and commenting on the manuscript. The work of Balázs Ráth is partially supported by OTKA (Hungarian National Research Fund) grant K100473, the Postdoctoral Fellowship of the Hungarian Academy of Sciences and the Bolyai Research Scholarship of the Hungarian Academy of Sciences.

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Correspondence to Balázs Ráth.

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Popov, S., Ráth, B. On Decoupling Inequalities and Percolation of Excursion Sets of the Gaussian Free Field. J Stat Phys 159, 312–320 (2015). https://doi.org/10.1007/s10955-015-1187-z

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  • DOI: https://doi.org/10.1007/s10955-015-1187-z

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