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作者：Dumitriu, Ioana; Pal, Soumik
作者单位：University of Washington; University of Washington Seattle
摘要：We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized.
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作者：Alsmeyer, Gerold; Biggins, J. D.; Meiners, Matthias
作者单位：University of Munster; University of Sheffield; Uppsala University
摘要：Given a sequence T = (T-i)(i >= 1) of nonnegative random variables, a function f on the positive halfline can be transformed to E Pi(i >= 1) f (tT(i)). We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Lapl...
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作者：Menshikov, Mikhail; Popov, Serguei; Ramirez, Alejandro F.; Vachkovskaia, Marina
作者单位：Durham University; Universidade Estadual de Campinas; Pontificia Universidad Catolica de Chile
摘要：In this paper we study a substantial generalization of the model of excited random walk introduced in [Electron. Commun. Probab. 8 (2003) 86-92] by Benjamini and Wilson. We consider a discrete-time stochastic process (X-n, n = 0, 1, 2, ...) taking values on Z(d), d >= 2, described as follows: when the particle visits a site for the first time, it has a uniformly-positive drift in a given direction l; when the particle is at a site which was already visited before, it has zero drift. Assuming u...
