Maximum of a log-correlated Gaussian field

Abstract. We study the maximum of a Gaussian field on [0, 1]d (d ≥ 1) whose correlations decay loga-rithmically with the distance. Kahane [22] introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas [19] [20] extended Kahane’s construction to the critical case and established the KPZ formula at criticality. Moreover, they made in [19] several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in [19]: we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale.