Results 1  10
of
51
Noise sensitivity of Boolean functions and applications to percolation, Inst. Hautes Études
, 1999
"... It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority ..."
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Cited by 73 (15 self)
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It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noisestable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on an n + 1 by n grid. A configuration is a function that assigns to every edge the value 0 or 1. Let ω be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges e with ω(e) = 1. By duality, the probability for having a crossing is 1/2. Fix an ǫ ∈ (0,1). For each edge e, let ω ′ (e) = ω(e) with probability 1 − ǫ, and ω ′ (e) = 1 − ω(e)
SpaceTime Structures from IIB Matrix Model, Prog
, 1998
"... We derive a long distance effective action for spacetime coordinates from a IIB matrix model. It provides us an effective tool to study the structures of spacetime. We prove the finiteness of the theory for finite N to all orders of the perturbation theory. Spacetime is shown to be inseparable and ..."
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Cited by 54 (3 self)
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We derive a long distance effective action for spacetime coordinates from a IIB matrix model. It provides us an effective tool to study the structures of spacetime. We prove the finiteness of the theory for finite N to all orders of the perturbation theory. Spacetime is shown to be inseparable and its dimensionality is dynamically determined. The IIB matrix model contains a mechanism to ensure the vanishing cosmological constant which does not rely on the manifest supersymmetry. We discuss possible mechanisms to obtain realistic dimensionality and gauge groups from the IIB matrix model.
Optimal Coding and Sampling of Triangulations
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
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Cited by 37 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
Discrete approaches to quantum gravity in four dimensions
 LIVING REVIEWS IN RELATIVITY
, 1998
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Rotational symmetry breaking in multimatrix models”, Phys
 Rev. D
"... Dedicated to the memory of João D. Correia We consider a class of multimatrix models with an action which is O(D)invariant, where D is the number of N × N Hermitian matrices Xµ, µ = 1,...,D. The action is a function of all the elementary symmetric functions of the matrix Tµν = Tr(XµXν)/N. We addre ..."
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Cited by 17 (3 self)
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Dedicated to the memory of João D. Correia We consider a class of multimatrix models with an action which is O(D)invariant, where D is the number of N × N Hermitian matrices Xµ, µ = 1,...,D. The action is a function of all the elementary symmetric functions of the matrix Tµν = Tr(XµXν)/N. We address the issue whether the O(D) symmetry is spontaneously broken when the size N of the matrices goes to infinity. The phase diagram in the space of the parameters of the model reveals the existence of a critical boundary where the O(D) symmetry is maximally broken. OUTP0229P
Continuous Family Of Invariant Subspaces For RDiagonal Operators
 Invent. Math
, 2001
"... We show that every R{diagonal operator x has a continuous family of invariant subspaces relative to the von Neumann algebra generated by x. This allows us to nd the Brown measure of x and to nd a new conceptual proof that Voiculescu's S{transform is multiplicative. Our considerations base on a new c ..."
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Cited by 16 (0 self)
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We show that every R{diagonal operator x has a continuous family of invariant subspaces relative to the von Neumann algebra generated by x. This allows us to nd the Brown measure of x and to nd a new conceptual proof that Voiculescu's S{transform is multiplicative. Our considerations base on a new concept of R{diagonality with amalgamation, for which we give several equivalent characterizations. 1.
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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Cited by 6 (0 self)
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
Critical and multicritical semirandom (1 + d)dimensional lattices and hard objects in d dimensions
 J. Phys. A Math. Gen
, 2002
"... We investigate models of (1+d)D Lorentzian semirandom lattices with one random (spacelike) direction and d regular (timelike) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an ex ..."
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Cited by 6 (2 self)
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We investigate models of (1+d)D Lorentzian semirandom lattices with one random (spacelike) direction and d regular (timelike) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an exact solution of a variety of new models including critical and multicritical generalized (1+1)D Lorentzian surfaces, with fractal dimensions dF = k+1, k = 1, 2, 3,..., as well as a new model of (1+2)D critical tetrahedral complexes, with fractal dimension dF = 12/5. Critical exponents and universal scaling functions follow from this solution. We finally establish a general connection between (1+d)D Lorentzian lattices and directedsite lattice animals in (1 + d) dimensions.
Dual string from lattice YangMills theory
"... Abstract. We review properties of lowerdimension vacuum defects observed in lattice simulations of SU(2) YangMills theories. One and twodimensional defects are associated with ultraviolet divergent action. The action is the same divergent as in perturbation theory but the fluctuations extend ove ..."
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Cited by 5 (2 self)
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Abstract. We review properties of lowerdimension vacuum defects observed in lattice simulations of SU(2) YangMills theories. One and twodimensional defects are associated with ultraviolet divergent action. The action is the same divergent as in perturbation theory but the fluctuations extend over submanifolds of the whole 4d space. The action is self tuned to a divergent entropy and the 2d defects can be thought of as dual strings populated with particles. The newly emerging 3d defects are closely related to the confinement mechanism. Namely, there is a kind of holography so that information on the confinement is encoded in a 3d submanifold. We introduce an SU(2) invariant classification scheme which allows for a unified description of d = 1,2,3 defects. The scheme fits known data and predicts that 3d defects are related to chiral symmetry breaking. Relation to stochastic vacuum model is briefly discussed as well.