Nicola Kistler

Poissonian statistics in the extremal process of branching Brownian motion

As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time

On a nonhierarchical version of the generalized random energy model

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A Quenched Large Deviation Principle and a Parisi Formula for a Perceptron Version of the GREM

, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time

Derrida’s Random Energy Models

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An ergodic theorem for the extremal process of branching Brownian motion

. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process obtained by retaining only those extremal particles which are also maximal inside the clusters converges in the limit of large

Small Perturbations of a Spin Glass System

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Universal structures in some mean field spin glasses and an application

to a random shift of a Poisson point process with exponential density. The last section discusses the Tidal Wave Conjecture by Lalley and Sellke [Ann. Probab. 15 (1987) 1052-1061] on the full limiting extremal process and its relation to the work of Chauvin and Rouault [Math. Nachr. 149 (1990) 41-59] on branching Brownian motion with atypical displacement.

Microcanonical Analysis of the Random Energy Model in a Random Magnetic Field

We introduce a natural nonhierarchical version of Derrida’s generalized random energy model. We prove that, in the thermodynamical limit, the free energy is the same as that of a suitably constructed GREM.

The extremal process of branching Brownian motion

We introduce a perceptron version of the Generalized Random Energy Model, and prove a quenched Sanov-type large deviation principle for the empirical distribution of the random energies. The dual of the rate function has a representation through a variational formula, which is closely related to the Parisi variational formula for the SK-model.

Genealogy of extremal particles of branching Brownian motion

We discuss Derrida’s random energy models under the light of the recent advances in the study of the extremes of highly correlated random fields. In particular, we present a refinement of the second moment method which provides a unifying approach to models where multiple scales can be identified, such is the case for e.g. branching diffusions, the 2-dim Gaussian free field, certain issues of percolation in high dimensions, or cover times. The method identifies some universal mechanisms which seemingly play a fundamental role also in the behavior of the extremes of the characteristic polynomials of certain random matrix ensembles, or in the extremes of the Riemann ζ-function along the critical line.