Publications (38)95.75 Total impact
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ABSTRACT: The bootstrap percolation (or threshold model) is a dynamic process modelling the propagation of an epidemic on a graph, where inactive vertices become active if their number of active neighbours reach some threshold. We study an optimization problem related to it, namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics, with and without a constraint on the time needed for the complete activation. This problem encompasses in special cases many extremal characteristics of graphs like their independence, decycling or domination number, and can also be seen as a packing problem of repulsive particles. We use the cavity method (including the effects of replica symmetry breaking), an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years. We have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs, the most striking being that 5regular random graph with a threshold of activation of 3 (resp. 6regular with threshold 4) have contagious sets containing a fraction 1/6 (resp. 1/4) of the total number of vertices. Equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5regular (resp. 6regular) random graph to destroy its 3core. We also investigated Survey Propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs.  [Show abstract] [Hide abstract]
ABSTRACT: We study various annealing dynamics, both classical and quantum, for simple meanfield models and explain how to describe their behavior in the thermodynamic limit in terms of differential equations. In particular we emphasize the differences between quantum annealing (i.e. evolution with Schr\"odinger equation) and simulated quantum annealing (i.e. annealing of a Quantum Monte Carlo simulation).  [Show abstract] [Hide abstract]
ABSTRACT: We present a study of the coloring problem (antiferromagnetic Potts model) of random regular graphs, submitted to quantum fluctuations induced by a transverse field, using the quantum cavity method and quantum MonteCarlo simulations. We determine the order of the quantum phase transition encountered at low temperature as a function of the transverse field and discuss the structure of the quantum spin glass phase. In particular, we conclude that the quantum adiabatic algorithm would fail to solve efficiently typical instances of these problems because of avoided level crossings within the quantum spin glass phase, caused by a competition between energetic and entropic effects.  [Show abstract] [Hide abstract]
ABSTRACT: Among various algorithms designed to exploit the specific properties of quantum computers with respect to classical ones, the quantum adiabatic algorithm is a versatile proposition to find the minimal value of an arbitrary cost function (ground state energy). Random optimization problems provide a natural testbed to compare its efficiency with that of classical algorithms. These problems correspond to mean field spin glasses that have been extensively studied in the classical case. This paper reviews recent analytical works that extended these studies to incorporate the effect of quantum fluctuations, and presents also some original results in this direction.  [Show abstract] [Hide abstract]
ABSTRACT: This paper deals with fullyconnected meanfield models of quantum spins with pbody ferromagnetic interactions and a transverse field. For p=2 this corresponds to the quantum CurieWeiss model (a special case of the LipkinMeshkovGlick model) which exhibits a secondorder phase transition, while for p>2 the transition is first order. We provide a refined analytical description both of the static and of the dynamic properties of these models. In particular we obtain analytically the exponential rate of decay of the gap at the firstorder transition. We also study the slow annealing from the pure transverse field to the pure ferromagnet (and vice versa) and discuss the effect of the firstorder transition and of the spinodal limit of metastability on the residual excitation energy, both for finite and exponentially divergent annealing times. In the quantum computation perspective this quantity would assess the efficiency of the quantum adiabatic procedure as an approximation algorithm.  [Show abstract] [Hide abstract]
ABSTRACT: The density of states of disordered hopping models generically exhibits an essential singularity around the edges of its support, known as a Lifshitz tail. We study this phenomenon on the Bethe lattice, i.e. for the largesize limit of random regular graphs, converging locally to the infinite regular tree, for both diagonal and offdiagonal disorder. The exponential growth of the volume and surface of balls on these lattices is an obstacle for the techniques used to characterize the Lifshitz tails in the finitedimensional case. We circumvent this difficulty by computing bounds on the moments of the density of states, and by deriving their implications on the behavior of the integrated density of states. 
Article: Quantum BiroliMézard model: Glass transition and superfluidity in a quantum lattice glass model
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ABSTRACT: We study the quantum version of a lattice model whose classical counterpart captures the physics of structural glasses. We discuss the role of quantum fluctuations in such systems and in particular their interplay with the amorphous order developed in the glass phase. We show that quantum fluctuations might facilitate the formation of the glass at low enough temperature. We also show that the glass transition becomes a firstorder transition between a superfluid and an insulating glass at very low temperature, and is therefore accompanied by phase coexistence between superfluid and glassy regions.  [Show abstract] [Hide abstract]
ABSTRACT: We study the quantum version of a simplified model of optimization problems, where quantum fluctuations are introduced by a transverse field acting on the qubits. We find a complex lowenergy spectrum of the quantum Hamiltonian, characterized by an abrupt condensation transition and a continuum of level crossings as a function of the transverse field. We expect this complex structure to have deep consequences on the behavior of quantum algorithms attempting to find solutions to these problems.  [Show abstract] [Hide abstract]
ABSTRACT: The goal of this chapter is to review recent analytical results about the growth of a (static) correlation length in glassy systems, and the connection that can be made between this length scale and the equilibrium correlation time of its dynamics. The definition of such a length scale is first given in a generic setting, including finitedimensional models, along with rigorous bounds linking it to the correlation time. We then present some particular cases (finite connectivity meanfield models, and Kac limit of finite dimensional systems) where this length can be actually computed. Comment: Chapter of "Dynamical heterogeneities in glasses, colloids, and granular media", Eds.: L. Berthier, G. Biroli, JP Bouchaud, L. Cipelletti and W. van Saarloos (Oxford University Press, to appear), more info at http://w3.lcvn.univmontp2.fr/~lucacip/DH_book.htm 
Article: FirstOrder Transitions and the Performance of Quantum Algorithms in Random Optimization Problems
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ABSTRACT: We present a study of the phase diagram of a random optimization problem in the presence of quantum fluctuations. Our main result is the characterization of the nature of the phase transition, which we find to be a firstorder quantum phase transition. We provide evidence that the gap vanishes exponentially with the system size at the transition. This indicates that the quantum adiabatic algorithm requires a time growing exponentially with system size to find the ground state of this problem. 
Article: Anderson Model on Bethe Lattices: Density of States, Localization Properties and Isolated Eigenvalue
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ABSTRACT: We revisit the Anderson localization problem on Bethe lattices, putting in contact various aspects which have been previously only discussed separately. For the case of connectivity 3 we compute by the cavity method the density of states and the evolution of the mobility edge with disorder. Furthermore, we show that below a certain critical value of the disorder the smallest eigenvalue remains delocalized and separated by all the others (localized) ones by a gap. We also study the evolution of the mobility edge at the center of the band with the connectivity, and discuss the large connectivity limit. 
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ABSTRACT: We introduce a version of the cavity method for diluted meanfield spin models that allows the computation of thermodynamic quantities similar to the FranzParisi quenched potential in sparse random graph models. This method is developed in the particular case of partially decimated random constraint satisfaction problems. This allows to develop a theoretical understanding of a class of algorithms for solving constraint satisfaction problems, in which elementary degrees of freedom are sequentially assigned according to the results of a message passing procedure (beliefpropagation). We confront this theoretical analysis to the results of extensive numerical simulations.  [Show abstract] [Hide abstract]
ABSTRACT: The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional selfconsistent equation. In this paper we obtain this equation for the BoseHubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent state path integral, leads in the large connectivity limit to the mean field treatment of Fisher et al. [Phys. Rev. B {\bf 40}, 546 (1989)] at the leading order, and to the bosonic Dynamical Mean Field Theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B {\bf 77}, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions. Comment: 27 pages, 16 figures, minor corrections 
Article: Connections to Statistical Physics
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ABSTRACT: This chapter surveys a part of the intense research activity that has been devoted by theoretical physicists to the study of randomly generated kSAT instances. It can be at first sight surprising that there is a connection between physics and computer science. However lowtemperature statistical mechanics concerns precisely the behaviour of the lowlying configurations of an energy landscape, in other words the optimization of a cost function. Moreover the ensemble of random kSAT instances exhibit phase transitions, a phenomenon mostly studied in physics (think for instance at the transition between liquid and gaseous water). Besides the introduction of general concepts of statistical mechanics and their translations in computer science language, the chapter presents results on the location of the satisfiability transition, the detailed picture of the satisfiable regime and the various phase transitions it undergoes, and algorithmic issues for random kSAT instances.  [Show abstract] [Hide abstract]
ABSTRACT: The cavity method is a well established technique for solving classical spin models on sparse random graphs (meanfield models with finite connectivity). Laumann et al. [arXiv:0706.4391] proposed recently an extension of this method to quantum spin1/2 models in a transverse field, using a discretized SuzukiTrotter imaginary time formalism. Here we show how to take analytically the continuous imaginary time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path integral representation of the imaginary time dynamics. As a side result we also show how this procedure can be used in simple heatbath like Monte Carlo simulations of generic quantum spin models. The replica symmetric continuous time quantum cavity method is formulated for a wide class of models, and applied as a simple example on the Bethe lattice ferromagnet in a transverse field. The results of the methods are confronted with various approximation schemes in this particular case. On this system we performed quantum Monte Carlo simulations that confirm the exactness of the cavity method in the thermodynamic limit. Comment: 25 pages, 15 figures, typos corrected  [Show abstract] [Hide abstract]
ABSTRACT: We review the connection between statistical mechanics and the analysis of random optimization problems, with particular emphasis on the random kSAT problem. We discuss and characterize the different phase transitions that are met in these problems, starting from basic concepts. We also discuss how statistical mechanics methods can be used to investigate the behavior of local search and decimation based algorithms.  [Show abstract] [Hide abstract]
ABSTRACT: We study the set of solutions of random ksatisfiability formulae through the cavity method. It is known that, for an interval of the clausetovariables ratio, this decomposes into an exponential number of pure states (clusters). We refine substantially this picture by: (i) determining the precise location of the clustering transition; (ii) uncovering a second `condensation' phase transition in the structure of the solution set for k larger or equal than 4. These results both follow from computing the large deviation rate of the internal entropy of pure states. From a technical point of view our main contributions are a simplified version of the cavity formalism for special values of the Parisi replica symmetry breaking parameter m (in particular for m=1 via a correspondence with the tree reconstruction problem) and new largek expansions. Comment: 30 pages, 14 figures, typos corrected, discussion of appendix C expanded with a new figure  [Show abstract] [Hide abstract]
ABSTRACT: Message passing algorithms have proved surprisingly successful in solving hard constraint satisfaction problems on sparse random graphs. In such applications, variables are fixed sequentially to satisfy the constraints. Message passing is run after each step. Its outcome provides an heuristic to make choices at next step. This approach has been referred to as `decimation,' with reference to analogous procedures in statistical physics. The behavior of decimation procedures is poorly understood. Here we consider a simple randomized decimation algorithm based on belief propagation (BP), and analyze its behavior on random ksatisfiability formulae. In particular, we propose a tree model for its analysis and we conjecture that it provides asymptotically exact predictions in the limit of large instances. This conjecture is confirmed by numerical simulations.  [Show abstract] [Hide abstract]
ABSTRACT: An instance of a random constraint satisfaction problem defines a random subset (the set of solutions) of a large product space chiN (the set of assignments). We consider two prototypical problem ensembles (random ksatisfiability and qcoloring of random regular graphs) and study the uniform measure with support on S. As the number of constraints per variable increases, this measure first decomposes into an exponential number of pure states ("clusters") and subsequently condensates over the largest such states. Above the condensation point, the mass carried by the n largest states follows a PoissonDirichlet process. For typical large instances, the two transitions are sharp. We determine their precise location. Further, we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem. The degree of correlation naturally affects the performances of many search/sampling algorithms. Empirical evidence suggests that local Monte Carlo Markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point. Finally, refined message passing techniques (such as survey propagation) may also beat this threshold.
Publication Stats
1k  Citations  
95.75  Total Impact Points  
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Institutions

20092014

Etablissement Français du Sang Alsace
Strasburg, Alsace, France


20082013

Polytech ParisUPMC
Lutetia Parisorum, ÎledeFrance, France


20012012

Ecole Normale Supérieure de Paris
 Laboratoire de Physique Théorique
Lutetia Parisorum, ÎledeFrance, France


20032011

French National Centre for Scientific Research
 Laboratoire Statistique et Génome
Lutetia Parisorum, ÎledeFrance, France


20072009

Pierre and Marie Curie University  Paris 6
 Laboratoire de Physique Théorique ENS (LPTENS)
Lutetia Parisorum, ÎledeFrance, France


20052008

Sapienza University of Rome
 Department of Physics
Roma, Latium, Italy
