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作者：Fountoulakis, Nikolaos; Kang, Mihyun; Koch, Christoph; Makai, Tamas
作者单位：University of Birmingham; Graz University of Technology; University of Warwick
摘要：A bootstrap percolation process on a graph with infection threshold r >= 1 is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least r infected neighbours becomes infected and remains so forever. Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in Z(d) that are caused ...
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作者：Hiraoka, Yasuaki; Shirai, Tomoyuki; Khanh Duy Trinh
作者单位：Tohoku University; Kyushu University
摘要：The persistent homology of a stationary point process on R-N is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for ...
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作者：Forman, Noah; Pal, Soumik; Rizzolo, Douglas; Winkel, Matthias
作者单位：University of Washington; University of Washington Seattle; University of Delaware; University of Oxford
摘要：We establish two results about local times of spectrally positive stable processes. The first is a general approximation result, uniform in space and on compact time intervals, in a model where each jump of the stable process may be marked by a random path. The second gives moment control on the Holder constant of the local times, uniformly across a compact spatial interval and in certain random time intervals. For the latter, we introduce the notion of a Levy process restricted to a compact i...
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作者：Roberts, Matthew, I; Sengul, Bati
作者单位：University of Bath; Bank of America Corporation
摘要：In this paper, we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs (G(t) : t is an element of [0, 1]), where initially we start with a critical Erdos-Renyi graph ER(n, 1/n), and then evolve forward in time by resampling each edge independently at rate 1. We show that the size of the largest connected component that appears during the time interval [0, 1] is of order n(2/3) log(1/3)n with high probability. This is in con...
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作者：Mendelson, Shahar; Rauhut, Holger; Ward, Rachel
作者单位：Technion Israel Institute of Technology; Australian National University; RWTH Aachen University; University of Texas System; University of Texas Austin
摘要：We study the recovery of sparse vectors from subsampled random convolutions via l(1)-minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a sub-Gaussian generator with independent entries, we improve previously known estimates: if the sparsity s is small enough, that is, s less than or similar to root n/log(n), we show that m greater than or similar to s log(en/s) measurements are sufficient to recover s-sparse v...
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作者：Edelsbrunner, Herbert; Nikitenko, Anton
作者单位：Institute of Science & Technology - Austria
摘要：Using the geodesic distance on the n-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson-Delaunay mosaic in n-dimensional Euclidean space. Assuming the points are not conta...
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作者：Ding, Xiucai; Yang, Fan
作者单位：University of Toronto; University of Wisconsin System; University of Wisconsin Madison
摘要：In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form Q = TX(TX)*, where X is an M-2 x N random matrix with X-ij = N-1/2 qij such that qij are i.i.d. random variables with zero mean and unit variance, and T is an M-1 x M-2 deterministic matrix such that T*T is diagonal. We study the asymptotic behavior of the largest eigenvalues of Q when M := min{M-1 x M-2...
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作者：Atar, Rami; Biswas, Anup; Kaspi, Haya; Ramanan, Kavita
作者单位：Technion Israel Institute of Technology; Indian Institute of Science Education & Research (IISER) Pune; Technion Israel Institute of Technology; Brown University
摘要：The Skorokhod map on the half-line has proved to be a useful tool for studying processes with nonnegativity constraints. In this work, we introduce a measure-valued analog of this map that transforms each element. of a certain class of cadlag paths that take values in the space of signed measures on [0, infinity) to a cadlag path that takes values in the space of nonnegative measures on [0, infinity) in such a way that for each x > 0, the path t (bar right arrow) zeta(t) [0, x] is transformed ...
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作者：Crane, Harry
作者单位：Rutgers University System; Rutgers University New Brunswick
摘要：Combinatorial Levy processes evolve on general state spaces of combinatorial structures, of which standard examples include processes on sets, graphs and n-ary relations and more general possibilities are given by processes on graphs with community structure and multilayer networks. In this setting, the usual Levy process properties of stationary, independent increments are defined in an unconventional way in terms of the symmetric difference operation on sets. The main theorems characterize b...
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作者：Chenavier, Nicolas; Robert, Christian Y.
作者单位：Universite du Littoral-Cote-d'Opale; Universite Claude Bernard Lyon 1
摘要：We consider the Voronoi tessellation based on a homogeneous Poisson point process in an Euclidean space. For a geometric characteristic of the cells (e.g., the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distributio...
