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328
From private attitude to public opinion: A dynamic theory of social impact
 Psychological Review
, 1990
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Nonequilibrium critical phenomena and phase transitions into absorbing states
 ADVANCES IN PHYSICS
, 2000
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Towards conformal invariance of 2D lattice models
 Proceedings of the international congress of mathematicians (ICM
"... Abstract. Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, selfavoiding polymers,...This has led to numerous exact (but nonrigorous) predictions of their scaling exponents and dimensions. We will discuss how to pro ..."
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Cited by 106 (10 self)
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Abstract. Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, selfavoiding polymers,...This has led to numerous exact (but nonrigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to SchrammLoewner Evolution.
A.: Differential equations for quantum correlation functions
 In: Proceedings of the Conference on YangBaxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory
, 1990
"... The quantum nonlinear Schrodinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest rveight vector permits us to represent correlation function as a determinant of a Fredholm integrai operator. This integral operator can be treated as the ..."
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Cited by 102 (9 self)
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The quantum nonlinear Schrodinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest rveight vector permits us to represent correlation function as a determinant of a Fredholm integrai operator. This integral operator can be treated as the GelfandLevitan operator for some new differentiai equation. These differential equations are written down in the paper. They generalize the fifth Painlive transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymptotics of quantum conelation functions. Quantum correlation function (Fredholm determinant) plays the role of z functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to r function of the nonlinear Schrodinger equation itself. For a penetrable Bose gas (finite coupling constant c) the correlator is rfunction of an integrodifferentiation equation.
Application of the τfunction theory of Painlevé equations to random matrices
 PV, PIII, the LUE, JUE and CUE
, 2002
"... Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidim ..."
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Cited by 76 (21 self)
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Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a nonnegative
Sparse Signal Recovery Using Markov Random Fields
"... Compressive Sensing (CS) combines sampling and compression into a single subNyquist linear measurement process for sparse and compressible signals. In this paper, we extend the theory of CS to include signals that are concisely represented in terms of a graphical model. In particular, we use Markov ..."
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Cited by 63 (13 self)
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Compressive Sensing (CS) combines sampling and compression into a single subNyquist linear measurement process for sparse and compressible signals. In this paper, we extend the theory of CS to include signals that are concisely represented in terms of a graphical model. In particular, we use Markov Random Fields (MRFs) to represent sparse signals whose nonzero coefficients are clustered. Our new modelbased recovery algorithm, dubbed Lattice Matching Pursuit (LaMP), stably recovers MRFmodeled signals using many fewer measurements and computations than the current stateoftheart algorithms. 1
The central limit theorem for local linear s tatistics in classical compact groups and related combinatorial identities
 Ann. Probab
, 2000
"... We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn. 1 ..."
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Cited by 53 (2 self)
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We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn. 1
Conformally invariant scaling limits: an overview and collection of open problems
 Proceedings of the International Congress of Mathematicians, Madrid (M. SanzSolé et
, 2007
"... Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions beco ..."
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Cited by 52 (2 self)
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Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a oneparameter family of random curves called Stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
Rigorous probabilistic analysis of equilibrium crystal shapes
 J. Math. Phys
, 2000
"... Abstract. The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results which have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom construct ..."
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Cited by 49 (18 self)
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Abstract. The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results which have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing
Statistical Mechanics, ThreeDimensionality and NPcompleteness I. Universality of Intractability for the Partition Function of the Ising Model Across NonPlanar Lattices (Extended Abstract)
"... This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working ..."
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Cited by 47 (1 self)
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This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working conjectures in the statistical mechanics literature, such as the CrossedBonds conjecture, and the impossibility to compute effectively the partition functions for any threedimensional lattice Ising model � these conjectures apply to the Onsager algebraic method, the Fermion operators method, and the combinatorial method based on Pfaffians. The fundamental results of the area, including those of Onsager, Kac, Feynman, Fisher, Kasteleyn, Temperley, Green, Hurst and more recently Barahona: for every Planar crystal lattice the partition functions for the nite sublattices can be computed in polynomialtime, paired with the results of this paper: for every NonPlanar crystal lattice computing the parition functions for the finite sublattices is NPcomplete, provide an exact characterization for several of the most studied Ising models. Our results settle at once, for several models, (1) the 2D nonplanar vs. 2D planar, (2) the nextnearest neighbour