Louis-Pierre Arguin

On the structure of quasi-stationary competing particle systems

We study point processes on the real line whose configurations X are locally finite, have a maximum and evolve through increments which are functions of correlated Gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {q ij ) i,j∈ ℕ. A probability measure on the pair (X, Q) is said to be quasi-stationary if the joint law of the gaps of X and of Q is invariant under the evolution. A known class of universally quasi-stationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchically nested Poisson-Dirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi-stationary. The main result of this work is a proof of this conjecture for the case where q ij assume only a finite number of values. The result is of relevance for mean-field spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchical organization of the Gibbs measure was first proposed as an ansatz.

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

POISSON-DIRICHLET STATISTICS FOR THE EXTREMES OF A LOG-CORRELATED GAUSSIAN FIELD

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Maximum of the Characteristic Polynomial of Random Unitary Matrices

, 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

Maxima of a randomized Riemann zeta function, and branching random walks

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Uniqueness of Ground States for Short-Range Spin Glasses in the Half-Plane

. 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

Non-unitary observables in the 2d critical Ising model

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Spin Glass Computations and Ruelle’s Probability Cascades

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.

Homology of Fortuin-Kasteleyn clusters of Potts models on the torus

We study the statistics of the extremes of a discrete Gaussian field with logarithmic correlations at the level of the Gibbs measure. The model is defined on the periodic interval [0,1]. It is based on a model introduced by Bacry and Muzy, and is similar to the logarithmic Random Energy Model studied by Carpentier and Le Doussal, and more recently by Fyodorov and Bouchaud. At low temperature, it is shown that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. In particular, this proves a conjecture of Carpentier and Le Doussal that the statistics of the extremes of the log-correlated field behave as those of i.i.d. Gaussian variables and of branching Brownian motion at the level of the Gibbs measure. The proof is based on the computation of the free energy of a perturbation of the model, where a scale-dependent variance is introduced, and on general tools of spin glass theory.

Fluctuation Bounds For Interface Free Energies in Spin Glasses

It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a