Showing papers by "Bálint Virág published in 2008"
TL;DR: In this paper, it was shown that the probability of having no eigenvalue in a fixed interval of size λ$ is given by \[\bigl(\ kappa_{\beta}+o(1)-bigr)\lambda^{\gamma_{β}}\exp\biggl(-{\bet a}{64}\lambda^2+\bigggl({\beta}{8}-{1}{4}\biggr)\lambda\biggr), where λ is an undetermined positive constant.
Abstract: We show that in the point process limit of the bulk eigenvalues of $\beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[\bigl(\ kappa_{\beta}+o(1)\bigr)\lambda^{\gamma_{\beta}}\exp\biggl(-{\bet a}{64}\lambda^2+\biggl({\beta}{8}-{1}{4}\biggr)\lambda\biggr)\] as $\lambda\to\infty$, where \[\gamma_{\beta}={1}{4}\biggl({\beta}{2}+{2}{\beta}-3\biggr)\] and $\kappa_{\beta}$ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157--165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron--Martin--Girsanov transformation in stochastic calculus.
34 citations
01 Jan 2008
TL;DR: In this paper, the authors consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth and show that the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise.
Abstract: We consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized.
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The graphs of the noise covariance structure for d = 4, 3, 2.1 from above.
2 citations
TL;DR: In this article, the authors consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth and show that the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise.
Abstract: We consider random walk on a mildly random environment on finite transitive d- regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized.
1 citations