13,572 research outputs found

    Functional summary statistics for point processes on the sphere with an application to determinantal point processes

    Full text link
    We study point processes on Sd\mathbb S^d, the dd-dimensional unit sphere Sd\mathbb S^d, considering both the isotropic and the anisotropic case, and focusing mostly on the spherical case d=2d=2. The first part studies reduced Palm distributions and functional summary statistics, including nearest neighbour functions, empty space functions, and Ripley's and inhomogeneous KK-functions. The second part partly discusses the appealing properties of determinantal point process (DPP) models on the sphere and partly considers the application of functional summary statistics to DPPs. In fact DPPs exhibit repulsiveness, but we also use them together with certain dependent thinnings when constructing point process models on the sphere with aggregation on the large scale and regularity on the small scale. We conclude with a discussion on future work on statistics for spatial point processes on the sphere

    Random point sets and their diffraction

    Full text link
    The diffraction of various random subsets of the integer lattice Zd\mathbb{Z}^{d}, such as the coin tossing and related systems, are well understood. Here, we go one important step beyond and consider random point sets in Rd\mathbb{R}^{d}. We present several systems with an effective stochastic interaction that still allow for explicit calculations of the autocorrelation and the diffraction measure. We concentrate on one-dimensional examples for illustrative purposes, and briefly indicate possible generalisations to higher dimensions. In particular, we discuss the stationary Poisson process in Rd\mathbb{R}^{d} and the renewal process on the line. The latter permits a unified approach to a rather large class of one-dimensional structures, including random tilings. Moreover, we present some stationary point processes that are derived from the classical random matrix ensembles as introduced in the pioneering work of Dyson and Ginibre. Their re-consideration from the diffraction point of view improves the intuition on systems with randomness and mixed spectra.Comment: 9 pages, 2 figures; talk presented at ICQ 11 (Sapporo

    A dynamical classification of the range of pair interactions

    Full text link
    We formalize a classification of pair interactions based on the convergence properties of the {\it forces} acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function (PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e., in the "usual" thermodynamic limit. For a pair interaction potential V(r) with V(r) \rightarrow \infty) \sim 1/r^a defining a {\it bounded} pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the {\it pair force} is absolutely integrable, i.e., for a > d-1, where d is the spatial dimension. We refer to this case as {\it dynamically short-range}, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood around it. For the {\it dynamically long-range} case, i.e., a \leq d-1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined in this limit. We discuss also how, for a \leq d-1 (and notably, for the case of gravity, a=d-2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant classification in this context is, however, that which divides pair forces with a > d-2 (or a < d-2), for which the PDF of the {\it difference in forces} is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as for the (marginal) case of gravity, be defined consistently in an infinite uniform system.Comment: 12 pages, 1 figure; significantly shortened and focussed, additional references, version to appear in J. Stat. Phy

    Piecewise Constant Martingales and Lazy Clocks

    Full text link
    This paper discusses the possibility to find and construct \textit{piecewise constant martingales}, that is, martingales with piecewise constant sample paths evolving in a connected subset of R\mathbb{R}. After a brief review of standard possible techniques, we propose a construction based on the sampling of latent martingales Z~\tilde{Z} with \textit{lazy clocks} θ\theta. These θ\theta are time-change processes staying in arrears of the true time but that can synchronize at random times to the real clock. This specific choice makes the resulting time-changed process Zt=Z~θtZ_t=\tilde{Z}_{\theta_t} a martingale (called a \textit{lazy martingale}) without any assumptions on Z~\tilde{Z}, and in most cases, the lazy clock θ\theta is adapted to the filtration of the lazy martingale ZZ. This would not be the case if the stochastic clock θ\theta could be ahead of the real clock, as typically the case using standard time-change processes. The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on (intervals of) R\mathbb{R}.Comment: 17 pages, 8 figure
    corecore