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Abstract
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We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices
near the cusp points of the eigenvalue density are universal. Together with the
companion paper by Erdős et al. (2018, arXiv:1809.03971), which proves the same
result for the complex Hermitian symmetry class, this completes the last remaining
case of the Wigner–Dyson–Mehta universality conjecture after bulk and edge
universalities have been established in the last years. We extend the recent
Dyson Brownian motion analysis at the edge by Landon and Yau (2017,
arXiv:1712.03881) to the cusp regime using the optimal local law by Erdős et
al. (2018, arXiv:1809.03971) and the accurate local shape analysis of the density by
Ajanki et al. (2015, arXiv:1506.05095) and Alt et al. (2018, arXiv:1804.07752). We
also present a novel PDE-based method to improve the estimate on eigenvalue
rigidity via the maximum principle of the heat flow related to the Dyson Brownian
motion.
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Keywords
cusp universality, Dyson Brownian motion, local law
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Mathematical Subject Classification 2010
Primary: 60B20, 15B52
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Milestones
Received: 28 January 2019
Revised: 17 June 2019
Accepted: 22 July 2019
Published: 12 October 2019
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