Corpus ID: 201671368

# The stochastic Airy operator at large temperature

@article{Dumaz2019TheSA,
title={The stochastic Airy operator at large temperature},
author={Laure Dumaz and Cyril Labb'e},
journal={arXiv: Probability},
year={2019}
}
• Published 29 August 2019
• Mathematics, Physics
• arXiv: Probability
It was shown in [J. A. Ramirez, B. Rider and B. Virag. J. Amer. Math. Soc. 24, 919-944 (2011)] that the edge of the spectrum of $\beta$ ensembles converges in the large $N$ limit to the bottom of the spectrum of the stochastic Airy operator. In the present paper, we obtain a complete description of the bottom of this spectrum when the temperature $1/\beta$ goes to $\infty$: we show that the point process of appropriately rescaled eigenvalues converges to a Poisson point process on $\mathbb{R… Expand 3 Citations #### Figures from this paper Poisson statistics for Gibbs measures at high temperature We consider a gas of N particles with a general two-body interaction and confined by an external potential in the mean field or high temperature regime, that is when the inverse temperature satisfiesExpand Operator level hard-to-soft transition for β-ensembles • Mathematics, Physics • 2020 The soft and hard edge scaling limits of$\beta$-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. It has been shown that by tuning the parameter of the hardExpand Strong approximation of Gaussian$\beta$-ensemble characteristic polynomials: the edge regime and the stochastic Airy function • Mathematics, Physics • 2020 We investigate the characteristic polynomials of the Gaussian$\beta$-ensemble for general$\beta>0$through its transfer matrix recurrence. We show that the rescaled characteristic polynomialExpand #### References SHOWING 1-10 OF 14 REFERENCES Tracy–Widom at High Temperature • Mathematics, Physics • 2014 We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature $$\beta$$β tends to $$0$$0. We prove that the minimal eigenvalue,Expand Localization of the continuous Anderson Hamiltonian in 1-D • Mathematics, Physics • Probability Theory and Related Fields • 2019 We study the bottom of the spectrum of the Anderson Hamiltonian $${\mathcal {H}}_L := -\partial _x^2 + \xi$$ H L : = - ∂ x 2 + ξ on [0, L ] driven by a white noise $$\xi$$ ξ and endowed withExpand Beta ensembles, stochastic Airy spectrum, and a diffusion • Mathematics, Physics • 2011 We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schroedinger operator -d^2/dx^2 + x +Expand Semigroups for One-Dimensional Schr\"odinger Operators with Multiplicative White Noise Let$ H:=-\tfrac12\Delta+V$be a one-dimensional continuum Schrodinger operator. Consider${\hat H}:= H+\xi$, where$\xi$is a Gaussian white noise. We prove that if the potential$V$is locallyExpand Poisson statistics at the edge of Gaussian beta-ensemble at high temperature • Cambyse Pakzad • Mathematics • Latin American Journal of Probability and Mathematical Statistics • 2019 We study the asymptotic edge statistics of the Gaussian$\beta$-ensemble, a collection of$n$particles, as the inverse temperature$\beta$tends to zero as$n$tends to infinity. In a certain decayExpand Gaussian Beta Ensembles at High Temperature: Eigenvalue Fluctuations and Bulk Statistics • Physics, Mathematics • Journal of Statistical Physics • 2018 We study the limiting behavior of Gaussian beta ensembles in the regime where $$\beta n = const$$βn=const as $$n \rightarrow \infty$$n→∞. The results are (1) Gaussian fluctuations for linearExpand A limit law for the ground state of Hill's equation AbstractIt is proved that the ground state Λ(L) of (−1)x the Schrödinger operator with white noise potential, on an interval of lengthL, subject to Neumann, periodic, or Dirichlet conditions,Expand Poisson Statistics for Matrix Ensembles at Large Temperature • Mathematics • 2015 In this article, we consider $$\beta$$β-ensembles, i.e. collections of particles with random positions on the real line having joint distribution$\$\begin{aligned} \frac{1}{Z_N(\beta )}\big |\DeltaExpand
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This paper constructs tridiagonal random matrix models for general (β>0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models forExpand