Local semicircle law in the bulk for Gaussian $β$-ensemble

Details Local semicircle law in the bulk for Gaussian $β$-ensemble Local semicircle law in the bulk for Gaussian $β$-ensemble

2011-12-09

arxiv.org/abs/1112.2016v2

Mathematics

Probability

26 pages, version 2 corrected typos

Scientific paper

We use the tridiagonal matrix representation to derive a local semicircle law for Gaussian beta ensembles at the optimal level of $n^{-1+\delta}$ for any $\delta > 0$. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of $n^{-1/2+\delta}$; then an induction argument allows us to reach the optimal level. This result was obtained in a different setting, using different methods, by Bourgade, Erd\"os, and Yau and in Bao and Su. Our approach is new and extends to other tridiagonal models, in particular, our approach does not use the commonly used "master-loop equation" and does not assume convexity nor analyticity on the potential but it does require a tridiagonal formulation.

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