Universiteit Utrecht

Department of Mathematics

Extremes and emergence of hierarchies in high-dimensional Gaussian random fields

Anton Klymovskiy
(EURandom, TU Eindhoven)

6 May 2010

We report on rigorous proofs of some of the heuristic results of Fyodorov & Sommers (2007) and Fyodorov & Bouchaud (2008) on spontaneous emergence of hierarchical structures in arbitrary high-dimensional Gaussian random fields with isotropic increments. In particular, we prove an exact (and computable) variational formula for the expected values of the extremes of such Gaussian random fields. Our proofs rely on spin-glass techniques of comparison with carefully chosen hierarchically correlated fields and on some remarkable properties of such probabilistic objects as Ruelle's probability cascades and the Bolthausen-Sznitman coalescent.

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