In this paper, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (CβE). More precisely, if X n is this characteristic polynomial and U the unit circle, we prove that:as well as an analogous statement for the imaginary part. The notation O(1) means that the corresponding family of random variables, indexed by n, is tight. This answers a conjecture of Fyodorov, Hiary and Keating, originally formulated for the β = 2 case, which corresponds to the CUE field.

Abstract. Considering a critical branching random walk on the real line. In a recent paper, Aïdékon [2] developed a powerful method to obtain the convergence in law of its minimum after a log-factor translation. By an adaptation of this method, we show that the point process formed by the branching random walk seen from the minimum converges in law to a decorated Poisson point process. This result, confirming a conjecture of Brunet and Derrida [10], can be viewed as a discrete analog of the corresponding results for the branching brownian motion, previously established by Arguin et al. [5] [6] and Aïdékon et al. [3].

Abstract. We study the maximum of a Gaussian field on [0,1] d (d ≥ 1) whose correlations decay logarithmically 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.

In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the so-called glassy phase. The limiting Gibbs weights are integrated atomic random measures with random intensity expressed in terms of the critical Gaussian multiplicative chaos constructed in [10,11]. This could be seen as a first rigorous step in the renormalization theory of super-critical Gaussian multiplicative chaos.