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作者：Chatterjee, Shirshendu; Zeitouni, Ofer
作者单位：City University of New York (CUNY) System; City College of New York (CUNY); Weizmann Institute of Science; New York University
摘要：We consider the searching for a trail in a maze composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side n based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero mean off the anomalous path and mean mu(n) on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. Arias-Castro et al. [Ann. Statis...
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作者：Foondun, Mohammud; Joseph, Mathew; Li, Shiu-Tang
作者单位：University of Strathclyde; University of Sheffield; Utah System of Higher Education; University of Utah
摘要：We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations. As a consequence, we prove various comparison principles extending earlier works of [Stoch. Stoch. Rep. 37 (1991) 225-245] and [Ann. Probab. 45 (2017) 377403] among others. Among other things, our results enable us to obtain sharp estimates on the moments of the solution. A main technical ingredient of our method is a local limit theorem which is of independe...
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作者：Gulikers, Lennart; Lelarge, Marc; Massoulie, Laurent
作者单位：Institut Polytechnique de Paris; Ecole Polytechnique; Universite PSL; Ecole Normale Superieure (ENS)
摘要：We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on n nodes, having i.i.d. weights (phi u)(u)(n)=i (possibly heavytailed), partitioned into q >= 2 asymptotically equal-sized clusters. The model parameters are two constants a, b > 0 and the finite second moment of the weights Phi((2)). Vertices u and v are connected by an edge with probability phi u phi v/na when they are in the same class and with probability phi u phi v/nb otherwise. We prove that it is informa...
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作者：Damron, Michael; Hanson, Jack; Lam, Wai-Kit
作者单位：University System of Georgia; Georgia Institute of Technology; City University of New York (CUNY) System; City College of New York (CUNY); Indiana University System; Indiana University Bloomington
摘要：First-passage percolation is a random growth model defined using i.i.d. edge-weights (t(e)) on the nearest-neighbor edges of Z(d). An initial infection occupies the origin and spreads along the edges, taking time t(e) to cross the edge e. In this paper, we study the size of the boundary of the infected (wet) region at time t, B(t). It is known that B(t) grows linearly, so its boundary partial derivative B(t) has size between ct(d-1) and Ct(d). Under a weak moment condition on the weights, we s...
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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：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：Beskos, Alexandros; Roberts, Gareth; Thiery, Alexandre; Pillai, Natesh
作者单位：University of London; University College London; University of Warwick; National University of Singapore; Harvard University
摘要：We study the asymptotic behaviour of the Random Walk Metropolis algorithm on ridged probability densities where most of the probability mass is distributed along some key directions. Such class of probability measures arise in various applied contexts including for instance Bayesian inverse problems where the posterior measure concentrates on a manifold when the noise variance goes to zero. When the target measure concentrates on a linear manifold, we derive analytically a diffusion limit for ...
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作者：Cattiaux, Patrick; Delebecque, Fanny; Pedeches, Laure
作者单位：Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Institut National des Sciences Appliquees de Toulouse
摘要：In this paper we revisit and generalize various stochastic models extending the deterministic Cucker-Smale model for self-organization. We study flocking and swarming properties. We show how these properties strongly depend on the structure and on the variance of the noise.
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作者：Goldsheid, Ilya; Sodin, Sasha
作者单位：University of London; Queen Mary University London; Tel Aviv University
摘要：The eigenvalues of the Hatano-Nelson non-Hermitian Anderson matrices, in the spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity parameter, are shown to be real and exponentially close to the Hermitian eigenvalues. This complements previous results, according to which the eigenvalues in the spectral regions in which the non-Hermiticity parameter exceeds the Lyapunov exponent are aligned on curves in the complex plane.
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作者：Li, Xiaoou; Liu, Jingchen; Lu, Jianfeng; Zhou, Xiang
作者单位：University of Minnesota System; University of Minnesota Twin Cities; Columbia University; Duke University; Duke University; City University of Hong Kong
摘要：Partial differential equations with random inputs have become popular models to characterize physical systems with uncertainty coming from imprecise measurement and intrinsic randomness. In this paper, we perform asymptotic rare-event analysis for such elliptic PDEs with random inputs. In particular, we consider the asymptotic regime that the noise level converges to zero suggesting that the system uncertainty is low, but does exist. We develop sharp approximations of the probability of a larg...
