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We study the accuracy of the expected Euler characteristic approximation to the distribution of the maximum of a smooth, centered , unit variance Gaussian process f. Using a point process representation of the error, valid for arbitrary smooth processes, we show that the error is in general exponentially smaller than any of the terms in the approximation.(More)
RANDOM FIELDS AND GEOMETRY published with Springer in 2007, but rather a companion volume, still under production, that gives a simpler version of the theory of the first book as well as lots of applications. You can find the original Random Fields and Geometry on the Springer site. Meanwhile, enjoy what is available of the second volume, and keep in mind(More)
We consider smooth, infinitely divisible random fields X(t), t ∈ M , M ⊂ R d , with regularly varying Lévy measure, and are interested in the geometric characteristics of the excursion sets Au = t ∈ M : X(t) > u over high levels u. For a large class of such random fields we compute the u → ∞ asymptotic joint distribution of the numbers of critical points,(More)
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the(More)
The explosion of the World Wide Web as a medium for information dissemination has made it important to understand its characteristics, in particular the distribution of its le sizes. This paper presents evidence that a number of le size distributions in the Web exhibit heavy tails, including les requested by users, les transmitted through the network,(More)