Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian free field

Abstract : In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions.The method is based on showing that the model admits a one-step replica symmetry breaking in spin glass terminology.This implies Poisson-Dirichlet statistics by general spin glass arguments.In this note, this approach is used to prove Poisson-Dirichlet statistics for the two-dimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis.
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Louis-Pierre Arguin, Olivier Zindy. Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian free field. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20, pp.1-19. ⟨10.1214/EJP.v20-3077⟩. ⟨hal-01214647⟩

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