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Edge universality of beta ensembles. (English) Zbl 1306.82010
The first main result of the paper is the edge universality for the beta ensembles for any \(\beta\geq 1\) under the following assumptions: (1) the equilibrium density is supported on a finite interval and (2) the potential is \(\mathcal C^4\) with at least logarithmic growth at infinity, bounded from below by the second derivative, and regular in the sense of A. B. J. Kuijlaars and K. T. R. McLaughlin [Commun. Pure Appl. Math. 53, No. 6, 736–785 (2000; Zbl 1022.31001)] (which means that the equilibrium density is positive in the interior of the interval of support and vanishes like a square root at each of the endpoints of the interval). The proof is based on a certain rigidity estimate for locations of particles. This estimate, in addition, allows to extend existing results on the bulk universality for the beta ensembles from analytic to \(\mathcal C^4\) potentials. (The universality for correlation functions holds for all \(\beta>0\), and for gaps for all \(\beta\geq 1\).) Unlike the main result, the rigidity estimate holds for all \(\beta>0\). The restriction on \(\beta\) in the universality result is due to the use of the local Dyson Brownian motion, which is well defined only for \(\beta\geq 1\). The authors believe that this restriction can be removed, but do not pursue this question in the present paper. Around the same time, the edge universality of the beta ensembles for all \(\beta>0\) and convex polynomial potentials was proved in [M. Krishnapur, B. Rider and B. Virag, “Universality of the stochastic Airy operator”, Preprint, arXiv:1306.4832].
The second main result of the paper is the edge universality of generalized Wigner matrices for all symmetry classes under the assumption of subexponential decay of normalized matrix elements. This extends previous works of various authors on edge universality of Wigner matrices, which essentially relied on the fact that the variances of the matrix elements were identical, to generalized Wigner matrices (with non-identical variances). The edge universality of statistics of generalized Wigner matrices was shown in [L. Erdős et al., Adv. Math. 229, No. 3, 1435–1515 (2012; Zbl 1238.15017)], but it was not proved there that the statistics are independent of the variances. This is achieved in the present work.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
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