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As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between $2M$ points for some integer $M$. The role of matrices is now played by $2M$-dimensional hypercubic arrays, and the determinant is replaced by a suitable generalization of it to such arrays -- Cayleys first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization.
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We derive joint factorial moment identities for point processes with Papangelou intensities. Our proof simplifies previous approaches to related moment identities and includes the setting of Poisson point processes. Applications are given to random transformations of point processes and to their distribution invariance properties.
Let $Y_i,igeq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $mathcal {M}subset mathbb{R}^d$ and consider sums $sum_{i=1}^nxi(n^{1/m}Y_i,{n^{1/m}Y_j}_{j=1}^n)$, where $xi$ is a real valued function defined on pairs $(y,mathcal {Y})$, with $yin mathbb{R}^d$ and $mathcal {Y}subset mathbb{R}^d$ locally finite. Subject to $xi$ 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 $xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $mathcal {M}$. 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 ${Y_i}_{i=1}^n$.
Our first result states that the orthogonal and symplectic Bessel processes are rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceeds by an estimate of the variance of additive statistics in the spirit of Ghosh and Peres. Second, a sufficient condition for number rigidity of stationary Pfaffian processes, relying on the Kolmogorov criterion for interpolation of stationary processes and applicable, in particular, to pfaffian sine-processes, is given in terms of the asymptotics of the spectral measure for additive statistics.
The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian particle systems with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. We start with the special case of pure hard core repulsion in order to show how to treat hard cores in general.
Affine point processes are a class of simple point processes with self- and mutually-exciting properties, and they have found useful applications in several areas. In this paper, we obtain large-time asymptotic expansions in large deviations and refined central limit theorem for affine point processes, using the framework of mod-phi convergence. Our results extend the large-time limit theorems in [Zhang et al. 2015. Math. Oper. Res. 40(4), 797-819]. The resulting explicit approximations for large deviation probabilities and tail expectations can be used as an alternative to importance sampling Monte Carlo simulations. Numerical experiments illustrate our results.
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