Friedrich Götze

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We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume however that(More)
We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the(More)
S E R G E Y G . B O B KOV , F R I E D R I C H G OÈ T Z E 2 and CHRISTIAN HOUDREÂ 3 School of Mathematics, University of Minnesota, Minneapolis MN 55455, USA. E-mail: bobkov@math.umn.edu Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany. E-mail: goetze@mathematik.uni-bielefeld.de School of Mathematics, Georgia Institute of Technology,(More)
For d-dimensional irrational ellipsoids E with d ≥ 9 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order o(r d−2). The estimate refines an earlier authors' bound of order O(r d−2) which holds for arbitrary ellipsoids, and is optimal for rational ellipsoids. As an application we(More)
LetHN be the set of all N×N (complex) Hermitian matrices, and let trA = ∑N i=1 aii denotes the trace of a square matrix A = (aij) N i,j=1. HN is a real Hilbert space of dimension N with respect to the symmetric bilinear form (A,B) 7→ trAB. Let lN denotes the unique Lebesgue measure on HN which satisfies the relation lN(Q) = 1 for every cube Q ⊂ HN with(More)
We present and investigate a general model for inhomogeneous random digraphs with labeled vertices, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and its orientation. For this model the critical point for the emergence of a giant component is determined via a branching process(More)