Extreme gaps between eigenvalues of Wigner matrices

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This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries.

• The proof relies on the Erd{\H o}s-Schlein-Yau dynamic approach.
• We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle.
• This observable also provides a simple and unified proof of universality in the bulk and at the edge, which is quantitative.