The glassy phase of complex branching Brownian motion

In this paper, we study complex valued branching Brownian motion in the so-called glassy phase, or also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on complex Gaussian multiplicative chaos. We will show that the limiting partition function can be expressed as a product of a Gaussian random variable, mainly due to the windings of the phase, and a stable transform of the so called derivative martingale, mainly due to the clustering of the modulus. The proof relies on the fine description of the extremal process available in the branching Brownian motion context.Comment: 23 pages; added references and a few details in the proof

Maximum of a log-correlated Gaussian field

We study the maximum of a Gaussian field on [0,1]^\d (\d \geq 1) whose correlations decay logarithmically with the distance. Kahane \cite{Kah85} introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas \cite{DRSV12a} \cite{DRSV12b} extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in \cite{DRSV12a} several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in \cite{DRSV12a}: 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

Glassy phase and freezing of log-correlated Gaussian potentials

International audienceIn 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 \cite{Rnew7,Rnew12}. This could be seen as a first rigorous step in the renormalization theory of super-critical Gaussian multiplicative chaos

Semi-Lagrangian simulations of the diocotron instability

We consider a guiding center simulation on an annulus. We propose here to revisit this test case by using a classical semi-Lagrangian approach. First, we obtain the conservation of the electric energy and mass for some adapted boundary conditions. Then we recall the dispersion relation and discussions on diff erent boundary conditions are detailed. Finally, the semi-Lagrangian code is validated in the linear phase against analytical growth rates given by the dispersion relation. Also we have validated numerically the conservation of electric energy and mass. Numerical issues/diffi culties due to the change of geometry can be tackled in such a test case which thus may be viewed as a fi rst intermediate step between a classical guiding center simulation in a 2D cartesian mesh and a slab 4D drift kinetic simulation