Gérard Ben Arous

Weierstrass Institute for Applied Analysis and Stochastics, Berlín, Berlin, Germany

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Publications (57)70.35 Total impact

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    ABSTRACT: The speed v(β) of a β-biased random walk on a Galton-Watson tree without leaves is increasing for β ≥ 1160. © 2013 Wiley Periodicals, Inc.
    Communications on Pure and Applied Mathematics 01/2014; · 3.34 Impact Factor
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    ABSTRACT: We introduce a general model of trapping for random walks on graphs. We give the possible scaling limits of these "Randomly Trapped Random Walks" on Z. These scaling limits include the well known Fractional Kinetics process, the Fontes-Isopi-Newman singular diffusion as well as a new broad class we call Spatially Subordinated Brownian Motions. We give sufficient conditions for convergence and illustrate these on two important examples.
    02/2013;
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    ABSTRACT: We consider a biased random walk Xn on a Galton–Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending on the bias β, such that |Xn| is of order nγ. Denoting Δn the hitting time of level n, we prove that Δn/n1/γ is tight. Moreover, we show that Δn/n1/γ does not converge in law (at least for large values of β). We prove that along the sequences nλ(k) = ⌊λβγk⌋, Δn/n1/γ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton–Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.
    The Annals of Probability 01/2012; 40(1). · 1.38 Impact Factor
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    ABSTRACT: The speed $v(\beta)$ of a $\beta$-biased random walk on a Galton-Watson tree without leaves is increasing for $\beta \geq 717$.
    11/2011;
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    Antonio Auffinger, Gerard Ben Arous
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    ABSTRACT: We analyze the landscape of general smooth Gaussian functions on the sphere in dimension N, when N is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom energy landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at energy levels below the limiting ground state energy the mean number of local minima is exponentially large. These two scenarios should correspond to the distinction between one-step replica symmetry breaking and full replica-symmetric breaking of the physics literature on spin glasses. In the former, we find a new way to derive the asymptotic complexity function as a function of the 1RSB Parisi functional.
    10/2011;
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    Gerard Ben Arous, Yueyun Hu, Stefano Olla, Ofer Zeitouni
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    ABSTRACT: We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrary deep traps.
    Annales de l Institut Henri Poincaré Probabilités et Statistiques 06/2011; · 0.93 Impact Factor
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    Gerard Ben Arous, Alan Hammond
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    ABSTRACT: As a model of trapping by biased motion in random structure, we study the time taken for a biased random walk to return to the root of a subcritical Galton-Watson tree. We do so for trees in which these biases are randomly chosen, independently for distinct edges, according to a law that satisfies a logarithmic non-lattice condition. The mean return time of the walk is in essence given by the total conductance of the tree. We determine the asymptotic decay of this total conductance, finding it to have a pure power-law decay. In the case of the conductance associated to a single vertex at maximal depth in the tree, this asymptotic decay may be analysed by the classical defective renewal theorem, due to the non-lattice edge-bias assumption. However, the derivation of the decay for total conductance requires computing an additional constant multiple outside the power-law that allows for the contribution of all vertices close to the base of the tree. This computation entails a detailed study of a convenient decomposition of the tree, under conditioning on the tree having high total conductance. As such, our principal conclusion may be viewed as a development of renewal theory in the context of random environments. For randomly biased random walk on a supercritical Galton-Watson tree with positive extinction probability, our main results may be regarded as a description of the slowdown mechanism caused by the presence of subcritical trees adjacent to the backbone that may act as traps that detain the walker. Indeed, this conclusion is exploited in \cite{GerardAlan} to obtain a stable limiting law for walker displacement in such a tree.
    01/2011;
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    G. Ben Arous, O. Gun
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    ABSTRACT: We consider Random Hopping Time (RHT) dynamics of the Sherrington - Kirkpatrick (SK) model and p-spin models of spin glasses. For any of these models and for any inverse temperature we prove that, on time scales that are sub-exponential in the dimension, the properly scaled clock process (time-change process) of the dynamics converges to an extremal process. Moreover, on these time scales, the system exhibits aging like behavior which we called extremal aging. In other words, the dynamics of these models ages as the random energy model (REM) does. Hence, by extension, this confirms Bouchaud's REM-like trap model as a universal aging mechanism for a wide range of systems which, for the first time, includes the SK model.
    10/2010;
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    A. Auffinger, G. Ben Arous, J. Cerny
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    ABSTRACT: We give an asymptotic evaluation of the complexity of spherical p-spin spin-glass models via random matrix theory. This study enables us to obtain detailed information about the bottom of the energy landscape, including the absolute minimum (the ground state), the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our approach allows us to compute the related TAP-complexity and extend the results known in the physics literature. As an independent tool, we prove a LDP for the k-th largest eigenvalue of the GOE, extending the results of Ben Arous, Dembo and Guionnett (2001).
    03/2010;
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    Gérard Ben Arous, Ivan Corwin
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    ABSTRACT: We consider the family of two-sided Bernoulli initial conditions for TASEP which, as the left and right densities ($\rho_-,\rho_+$) are varied, give rise to shock waves and rarefaction fans---the two phenomena which are typical to TASEP. We provide a proof of Conjecture 7.1 of [Progr. Probab. 51 (2002) 185--204] which characterizes the order of and scaling functions for the fluctuations of the height function of two-sided TASEP in terms of the two densities $\rho_-,\rho_+$ and the speed $y$ around which the height is observed. In proving this theorem for TASEP, we also prove a fluctuation theorem for a class of corner growth processes with external sources, or equivalently for the last passage time in a directed last passage percolation model with two-sided boundary conditions: $\rho_-$ and $1-\rho_+$. We provide a complete characterization of the order of and the scaling functions for the fluctuations of this model's last passage time $L(N,M)$ as a function of three parameters: the two boundary/source rates $\rho_-$ and $1-\rho_+$, and the scaling ratio $\gamma^2=M/N$. The proof of this theorem draws on the results of [Comm. Math. Phys. 265 (2006) 1--44] and extensively on the work of [Ann. Probab. 33 (2005) 1643--1697] on finite rank perturbations of Wishart ensembles in random matrix theory.
    The Annals of Probability 05/2009; 39(2011). · 1.38 Impact Factor
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    G. Ben Arous, V. Kargin
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    ABSTRACT: We continue here the study of free extreme values begun in Ben Arous and Voiculescu (2006). We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure (Voiculescu (1998), Barndorff-Nielsen and Thorbjornsen (2005)). We relate this convergence to the free extremal laws introduced in Ben Arous and Voiculescu (2006) and give the limit laws for free order statistics.
    Probability Theory and Related Fields 04/2009; · 1.39 Impact Factor
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    Gérard Ben Arous, Alexey Kuptsov
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    ABSTRACT: We survey in this paper a universality phenomenon which shows that some characteristics of complex random energy landscapes are model-independent, or universal. This universality, called REM-universality, was discovered by S. Mertens and H. Bauke in the context of combinatorial optimization. We survey recent advances on the extent of this REM-universality for equilibrium as well as dynamical properties of spin glasses. We also focus on the limits of REM-universality, i.e., when it ceases to be valid. Mathematics Subject Classification (2000)82B44-82D30-82C44-60G15-60G55 KeywordsSpin glasses-random energy model-extreme values-Gaussian processes-statistical mechanics-disordered media
    12/2008: pages 45-84;
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    Gérard Ben Arous, Anton Bovier, Jiří Černý
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    ABSTRACT: We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β>0, there exists a constant γ β ,p >0, such that for all exponential time scales, exp(γ N), with γ < γ β ,p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β 2<1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.
    Communications in Mathematical Physics 08/2008; 282(3):663-695. · 1.97 Impact Factor
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    Gérard Ben Arous, Anton Bovier, Jiří Černý
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    ABSTRACT: Ageing has become the paradigm for describing dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of the effective dynamics of such systems. In this Letter we show that in a wide class of mean field models and on a wide range of time scales, ageing occurs precisely as predicted by the random energy model-like trap model of Bouchaud and Dean. This is the first rigorous result concerning ageing in mean field models other than the random energy model and the spherical model.
    Journal of Statistical Mechanics Theory and Experiment 04/2008; 2008(04):L04003. · 1.87 Impact Factor
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    Gérard Ben Arous, Jiří Černý
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    ABSTRACT: We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large, two-dimensional tori and for trap dynamics of the random energy model on a broad range of time scales. © 2006 Wiley Periodicals, Inc.
    Communications on Pure and Applied Mathematics 02/2008; 61(3):289 - 329. · 3.34 Impact Factor
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    Antonio Auffinger, Gerard Ben Arous, Sandrine Peche
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    ABSTRACT: We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix.
    Annales de l Institut Henri Poincaré Probabilités et Statistiques 11/2007; · 0.93 Impact Factor
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    Gerard Ben Arous, Alice Guionnet
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    ABSTRACT: Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, converges almost surely and in expectation to the so-called semicircular distribution as $N$ goes to infinity. In this paper we study the same question when $P$ is in the domain of attraction of an $\alpha$-stable law. We prove that if we renormalize the eigenvalues by a constant $a_N$ of order $N^{\frac{1}{\alpha}}$, the corresponding spectral distribution converges in expectation towards a law $\mu_\alpha$ which only depends on $\alpha$. We characterize $\mu_\alpha$ and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero.
    Communications in Mathematical Physics 08/2007; · 1.97 Impact Factor
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    Gerard Ben Arous, Veronique Gayrard, Alexey Kuptsov
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    ABSTRACT: We introduce here a new universality conjecture for levels of random Hamiltonians, in the same spirit as the local REM conjecture made by S. Mertens and H. Bauke. We establish our conjecture for a wide class of Gaussian and non-Gaussian Hamiltonians, which include the $p$-spin models, the Sherrington-Kirkpatrick model and the number partitioning problem. We prove that our universality result is optimal for the last two models by showing when this universality breaks down.
    01/2007;
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    Gérard Ben Arous, Jiří Černý
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    ABSTRACT: We give the “quenched” scaling limit of Bouchaud’s trap model in d≥2. This scaling limit is the fractional-kinetics process, that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator.
    The Annals of Probability 01/2007; 35(2007). · 1.38 Impact Factor
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    Gerard Ben Arous, Veronique Gayrard
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    ABSTRACT: This work addresses potential theoretic questions for the standard nearest neighbor random walk on the hypercube $\{-1,+1\}^N$. For a large class of subsets $A\subset\{-1,+1\}^N$ we give precise estimates for the harmonic measure of $A$, the mean hitting time of $A$, and the Laplace transform of this hitting time. In particular, we give precise sufficient conditions for the harmonic measure to be asymptotically uniform, and for the hitting time to be asymptotically exponentially distributed, as $N\to\infty$. Our approach relies on a $d$-dimensional extension of the Ehrenfest urn scheme called lumping and covers the case where $d$ is allowed to diverge with $N$ as long as $d\leq\alpha_0\frac{N}{\log N}$ for some constant $0<\alpha_0<1$.
    12/2006;

Publication Stats

770 Citations
70.35 Total Impact Points

Institutions

  • 2008
    • Weierstrass Institute for Applied Analysis and Stochastics
      Berlín, Berlin, Germany
    • CUNY Graduate Center
      New York City, New York, United States
  • 2006
    • Université Paris-Sud 11
      Orsay, Île-de-France, France
  • 1998–2005
    • École Polytechnique Fédérale de Lausanne
      Lausanne, Vaud, Switzerland
  • 1995–1997
    • French National Centre for Scientific Research
      Lutetia Parisorum, Île-de-France, France