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Mézard M, Zecchina R. Random K-satisfiability problem: from an analytic solution to an efficient algorithm. PHYSICAL REVIEW E 2002; 66:056126. [PMID: 12513575 DOI: 10.1103/physreve.66.056126] [Citation(s) in RCA: 300] [Impact Index Per Article: 13.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2002] [Indexed: 11/07/2022]
Abstract
We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the K=3 satisfiability problem, with very good performances.
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Müller M, Krzakala F, Mézard M. The secondary structure of RNA under tension. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2002; 9:67-77. [PMID: 15010932 DOI: 10.1140/epje/i2002-10057-5] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We study the force-induced unfolding of random disordered RNA or single-stranded DNA polymers. The system undergoes a second-order phase transition from a collapsed globular phase at low forces to an extensive necklace phase with a macroscopic end-to-end distance at high forces. At low temperatures, the sequence inhomogeneities modify the critical behaviour. We provide numerical evidence for the universality of the critical exponents which, by extrapolation of the scaling laws to zero force, contain useful information on the ground-state (f = 0) properties. This provides a good method for quantitative studies of scaling exponents characterizing the collapsed globule. In order to get rid of the blurring effect of thermal fluctuations, we restrict ourselves to the ground state at fixed external force. We analyze the statistics of rearrangements, in particular below the critical force, and point out its implications for force-extension experiments on single molecules.
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Mézard M, Parisi G, Zecchina R. Analytic and algorithmic solution of random satisfiability problems. Science 2002; 297:812-5. [PMID: 12089451 DOI: 10.1126/science.1073287] [Citation(s) in RCA: 229] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio alpha of clauses to variables less than a threshold alphac are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an intermediate phase below alphac, where the proliferation of metastable states is responsible for the onset of complexity in search algorithms. We introduce a class of optimization algorithms that can deal with these metastable states; one such algorithm has been tested successfully on the largest existing benchmark of K-satisfiability.
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Abstract
Motivated by the concept of geometrical frustration, we introduce a class of statistical mechanics lattice models for the glass transition. Monte Carlo simulations in three dimensions show that they display a dynamical glass transition which is very similar to that observed in other off-lattice systems and which does not depend on a specific dynamical rule. A mean-field study shows the existence of a discontinuous glass transition, in agreement with the numerical observations.
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Martin-Mayor V, Mézard M, Parisi G, Verrocchio P. The dynamical structure factor in topologically disordered systems. J Chem Phys 2001. [DOI: 10.1063/1.1349709] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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31
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Giardina I, Bouchaud JP, Mézard M. Proliferation assisted transport in a random environment. ACTA ACUST UNITED AC 2001. [DOI: 10.1088/0305-4470/34/16/104] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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32
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Montanari A, Mézard M. Hairpin formation and elongation of biomolecules. PHYSICAL REVIEW LETTERS 2001; 86:2178-2181. [PMID: 11289884 DOI: 10.1103/physrevlett.86.2178] [Citation(s) in RCA: 44] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2000] [Indexed: 05/23/2023]
Abstract
We introduce a model of thermalized conformations in space of RNA-or single stranded DNA-molecules, which includes the possibility of hairpin formation. This model contains the usual secondary structure information, but extends it to the study of one element of the ternary structure, namely the end-to-end distance. The computed force-elongation characteristics are in good agreement with some recent measurements on single stranded DNA molecules.
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Da Silva LF, Mézard M. [The Romeurope Project. The health of the Roms in a situation of total exclusion]. KRANKENPFLEGE. SOINS INFIRMIERS 2001; 94:56-9. [PMID: 11944406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [MESH Headings] [Subscribe] [Scholar Register] [Indexed: 02/24/2023]
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34
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Coluzzi B, Mézard M, Parisi G, Verrocchio P. Thermodynamics of binary mixture glasses. J Chem Phys 1999. [DOI: 10.1063/1.480246] [Citation(s) in RCA: 94] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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35
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Hazareesing A, Mézard M. Wandering of a contact line at thermal equilibrium. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:1269-78. [PMID: 11969885 DOI: 10.1103/physreve.60.1269] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/1998] [Indexed: 04/18/2023]
Abstract
We reconsider the problem of the solid-liquid-vapor contact-line on a disordered substrate, in the collective pinning regime. We perform a replica variational calculation which confirms the scaling behavior obtained from Larkin-Imry-Ma-like arguments and provides a quantitative prediction for the correlation function of the line. This prediction is in good agreement with experimental findings for the case of superfluid helium on a caesium substrate.
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Mézard M, Parisi G. A first-principle computation of the thermodynamics of glasses. J Chem Phys 1999. [DOI: 10.1063/1.479193] [Citation(s) in RCA: 187] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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37
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Barrat A, Burioni R, Mézard M. Dynamics within metastable states in a mean-field spin glass. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/29/5/001] [Citation(s) in RCA: 81] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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39
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40
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41
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42
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43
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Bouchaud JP, Mézard M. Velocity fluctuations in forced Burgers turbulence. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:5116-5121. [PMID: 9965691 DOI: 10.1103/physreve.54.5116] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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44
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Balents L, Bouchaud JP, Mézard M. The Large Scale Energy Landscape of Randomly Pinned Objects. ACTA ACUST UNITED AC 1996. [DOI: 10.1051/jp1:1996112] [Citation(s) in RCA: 58] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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45
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Bouchaud JP, Mézard M, Parisi G. Scaling and intermittency in Burgers turbulence. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:3656-3674. [PMID: 9963845 DOI: 10.1103/physreve.52.3656] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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46
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Mézard M, Monasson R. Glassy transition in the three-dimensional random-field Ising model. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:7199-7202. [PMID: 9974688 DOI: 10.1103/physrevb.50.7199] [Citation(s) in RCA: 42] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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48
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Bouchaud JP, Mézard M. Self induced quenched disorder: a model for the glass transition. ACTA ACUST UNITED AC 1994. [DOI: 10.1051/jp1:1994240] [Citation(s) in RCA: 98] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
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50
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Bouchaud JP, Mézard M, Yedidia JS. Variational theory for the pinning of vortex lattices by impurities. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:14686-14701. [PMID: 10003570 DOI: 10.1103/physrevb.46.14686] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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