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Register a new userRobustness of large-scale stochastic matrices to localized perturbations
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Added by
Giacomo Como
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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Upper bounds are derived on the total variation distance between the invariant distributions of two stochastic matrices differing on a subset W of rows. Such bounds depend on three parameters: the mixing time and the minimal expected hitting time on W for the Markov chain associated to one of the matrices; and the escape time from W for the Markov chain associated to the other matrix. These results, obtained through coupling techniques, prove particularly useful in scenarios where W is a small subset of the state space, even if the difference between the two matrices is not small in any norm. Several applications to large-scale network problems are discussed, including robustness of Googles PageRank algorithm, distributed averaging and consensus algorithms, and interacting particle systems.
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Extending our previous work on the robustness of inflation to perturbations in the scalar field, we investigate the effects of perturbations in the transverse traceless part of the extrinsic curvature on the evolution of an inhomogeneous inflaton field. Focusing on small field models, we show that these additional metric inhomogeneities initially reduce the total number of e-folds as the amplitude increases, but that the reduction saturates and even reverses above a certain amplitude. We present an argument that this is due to the presence of a large initial Hubble friction when metric perturbations are large.
The problem of reliability of a large distributed system is analyzed via a new mathematical model. A typical framework is a system where a set of files are duplicated on several data servers. When one of these servers breaks down, all copies of files stored on it are lost. In this way, repeated failures may lead to losses of files. The efficiency of such a network is directly related to the performances of the mechanism used to duplicate files on servers. In this paper we study the evolution of the network using a natural duplication policy giving priority to the files with the least number of copies. We investigate the asymptotic behavior of the network when the number $N$ of servers is large. The analysis is complicated by the large dimension of the state space of the empirical distribution of the state of the network. A stochastic model of the evolution of the network which has values in state space whose dimension does not depend on $N$ is introduced. Despite this description does not have the Markov property, it turns out that it is converging in distribution, when the number of nodes goes to infinity, to a nonlinear Markov process. The rate of decay of the network, which is the key characteristic of interest of these systems, can be expressed in terms of this asymptotic process. The corresponding mean-field convergence results are established. A lower bound on the exponential decay, with respect to time, of the fraction of the number of initial files with at least one copy is obtained.
A Freidlin-Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [3].
We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dysons Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies on cases where the initial density vanishes at an interior point of the support. We show that the time to reach universality becomes larger if the density vanishes faster or if the initial points show less rigidity.
We determine the operator limit for large powers of random tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_beta$ process, which describes the largest eigenvalues in the $beta$ ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Ram{i}rez, Rider, and Vir{a}g. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.
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