Journal article

The sharp phase transition for level set percolation of smooth planar Gaussian fields

S Muirhead, H Vanneuville

Annales de l'institut Henri Poincare (B) Probability and Statistics | Institute of Mathematical Statistics | Published : 2020


We prove that the connectivity of the level sets of a wide class of smooth centred planar Gaussian fields exhibits a phase transition at the zero level that is analogous to the phase transition in Bernoulli percolation. In addition to symmetry, positivity and regularity conditions, we assume only that correlations decay polynomially with exponent larger than two – roughly equivalent to the integrability of the covariance kernel – whereas previously the phase transition was only known in the case of the Bargmann–Fock covariance kernel which decays super-exponentially. We also prove that the phase transition is sharp, demonstrating, without any further assumption on the decay of correlations, ..

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University of Melbourne Researchers


Awarded by Engineering and Physical Sciences Research Council (EPSRC)

Awarded by ERC grant Liko

Funding Acknowledgements

The first author was supported by the Engineering and Physical Sciences Research Council (EPSRC) Grant EP/N009436/1 "The many faces of random characteristic polynomials". The second author was supported by the ERC grant Liko No. 676999.