Poisson statistics at the edge of Gaussian beta-ensembles at high temperature
TLDR
In this paper, the authors studied the edge statistics of the Gaussian ensemble and proved that the associated extreme point process converges in distribution to a Poisson point process as the inverse temperature tends to zero as the number of particles tends to infinity.Abstract:
We study the asymptotic edge statistics of the Gaussian $\beta$-ensemble, a collection of $n$ particles, as the inverse temperature $\beta$ tends to zero as $n$ tends to infinity. In a certain decay regime of $\beta$, the associated extreme point process is proved to converge in distribution to a Poisson point process as $n\to +\infty$. We also extend a well known result on Poisson limit for Gaussian extremes by showing the existence of an edge regime that we did not find in the literature.read more
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Poisson statistics for Gibbs measures at high temperature
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Poisson statistics for beta ensembles on the real line at high temperature
TL;DR: In this paper, the authors studied the local behavior of the beta ensembles on the real line in a high temperature regime and showed that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.
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Poisson statistics for beta ensembles on the real line at high temperature
TL;DR: In this article, the authors studied the local behavior of the beta ensembles on the real line in a high temperature regime and showed that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.
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The stochastic Airy operator at large temperature
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TL;DR: Ramirez et al. as mentioned in this paper showed that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on the edge of the spectrum of the stochastic Airy operator.
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Large Deviations Principle for the Largest Eigenvalue of the Gaussian $$\beta $$β-Ensemble at High Temperature
TL;DR: In this article, the largest particle satisfies a large deviations principle in the Gaussian ensemble when the number of particles scales with n such that it converges in probability to 2, the rightmost point of the semicircle law.
References
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Book
Extreme Values, Regular Variation, and Point Processes
TL;DR: In this paper, the authors present a survey of the main domains of attraction and norming constants in point processes and point processes, and their relationship with multivariate extremity processes.
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Matrix models for beta ensembles
Ioana Dumitriu,Alan Edelman +1 more
TL;DR: In this article, tridiagonal random matrix models for general (β>0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles were constructed.
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Beta ensembles, stochastic Airy spectrum, and a diffusion
TL;DR: In this paper, it was shown that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigen values of the random Schrodinger operator − d2 dx2 + x+ 2 √ β bx restricted to the positive half-line.
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Continuum limits of random matrices and the Brownian carousel
Benedek Valkó,Bálint Virág +1 more
TL;DR: The authors showed that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β, a translation invariant point process.