5,268 research outputs found

    Limit theory for point processes in manifolds

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    Let Yi,i1Y_i,i\geq1, be i.i.d. random variables having values in an mm-dimensional manifold MRd\mathcal {M}\subset \mathbb{R}^d and consider sums i=1nξ(n1/mYi,{n1/mYj}j=1n)\sum_{i=1}^n\xi(n^{1/m}Y_i,\{n^{1/m}Y_j\}_{j=1}^n), where ξ\xi is a real valued function defined on pairs (y,Y)(y,\mathcal {Y}), with yRdy\in \mathbb{R}^d and YRd\mathcal {Y}\subset \mathbb{R}^d locally finite. Subject to ξ\xi satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of ξ\xi on homogeneous Poisson point processes on mm-dimensional hyperplanes tangent to M\mathcal {M}. We apply the general results to establish the limit theory of dimension and volume content estimators, R\'{e}nyi and Shannon entropy estimators and clique counts in the Vietoris-Rips complex on {Yi}i=1n\{Y_i\}_{i=1}^n.Comment: Published in at http://dx.doi.org/10.1214/12-AAP897 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Growing Perfect Decagonal Quasicrystals by Local Rules

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    A local growth algorithm for a decagonal quasicrystal is presented. We show that a perfect Penrose tiling (PPT) layer can be grown on a decapod tiling layer by a three dimensional (3D) local rule growth. Once a PPT layer begins to form on the upper layer, successive 2D PPT layers can be added on top resulting in a perfect decagonal quasicrystalline structure in bulk with a point defect only on the bottom surface layer. Our growth rule shows that an ideal quasicrystal structure can be constructed by a local growth algorithm in 3D, contrary to the necessity of non-local information for a 2D PPT growth.Comment: 4pages, 2figure

    Mathematics of random growing interfaces

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    We establish a thermodynamic limit and Gaussian fluctuations for the height and surface width of the random interface formed by the deposition of particles on surfaces. The results hold for the standard ballistic deposition model as well as the surface relaxation model in the off-lattice setting. The results are proved with the aid of general limit theorems for stabilizing functionals of marked Poisson point processes.Comment: 12 page

    Gaussian limits for multidimensional random sequential packing at saturation (extended version)

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    Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume λ\lambda is asymptotically normal as λ\lambda \to \infty. We provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field. The method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization.Comment: 31 page

    The fields of uniformly accelerated charges in de Sitter spacetime

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    The scalar and electromagnetic fields of charges uniformly accelerated in de Sitter spacetime are constructed. They represent the generalization of the Born solutions describing fields of two particles with hyperbolic motion in flat spacetime. In the limit Lambda -> 0, the Born solutions are retrieved. Since in the de Sitter universe the infinities I^+- are spacelike, the radiative properties of the fields depend on the way in which a given point of I^+- is approached. The fields must involve both retarded and advanced effects: Purely retarded fields do not satisfy the constraints at the past infinity I^-.Comment: 5 pages, 3 figures, RevTeX; Slightly expanded version of the paper published in Physical Review Letters. (The published version can be generated from the same TeX source.); problem with the postscript fixe
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