The real Ginibre ensemble consists of square real matrices whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius of a real Ginibe matrix follows a different limiting law for purely real eigenvalues than for non-real ones. Building on previous work by Rider, Sinclair and Poplavskyi, Tribe, Zaboronski, we will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. This system is directly related to several of the most interesting nonlinear evolution equations in 1+1 dimensions which are solvable by the inverse scattering method, for instance the nonlinear Schr¨odinger equation. The results of this talk are based on the recent preprint arXiv:1808.02419, joint with Jinho Baik.[-]

La théorie des valeurs extrêmes décrit le comportement du maximum d'une suite de variables aléatoires i.i.d. à valeurs réelles. L'une des distributions limites possibles, la loi de Gumbel, apparaît également dans l'asymptotique en bruit faible du temps de transition réactive pour des équations différentielles stochastiques métastables. Nous décrivons des résultats récents en dimension 1 et leur interprétation, et donnons un résultat en dimension 2, motivé par le phénomène de synchronisation d'oscillateurs couplés.[-]

In this talk we discuss the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has in nite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. In the lighter-tailed case, we study the asymptotics of the scaled position of the rightmost particle in the n-th generation and show the existence of a non-trivial constant. This is a joint work with Souvik Ray (Stanford), Parthanil Roy (ISI, Bangalore) and Philippe Soulier (Universite Paris Nanterre).[-]

Statistical modelling of complex dependencies in extreme events requires meaningful sparsity structures in multivariate extremes. In this context two perspectives on conditional independence and graphical models have recently emerged: One that focuses on threshold exceedances and multivariate pareto distributions, and another that focuses on max-linear models and directed acyclic graphs. What connects these notions is the exponent measure that lies at the heart of each approach. In this work we develop a notion of conditional independence defined directly on the exponent measure (and even more generally on measures that explode at the origin) that extends recent work of Engelke and Hitz (2019), who had been confined to homogeneous measures with density. We prove easier checkable equivalent conditions to verify this new conditional independence in terms of a reduction to simple test classes, probability kernels and density factorizations. This provides a pathsway to graphical modelling among general multivariate (max-)infinitely distributions. Structural max-linear models turn out to form a Bayesian network with respect to our new form of conditional independence.[-]