The bulk eigenvalue asymptotic conjecture for Wigner matrices

Let $M_n$ be a Wigner matrix with atom variable $\eta$, let $\lambda_i$ denote its ordered eigenvalues, and let $\gamma_i$ be the corresponding semicircle quantiles. For fixed $\delta>0$ and indices satisfying $\delta n\leq i\leq (1-\delta)n$, there are a bounded quantity $C_{i,n}$ depending only on $i,n$ and an absolute constant $c>0$.

Bulk eigenvalue asymptotic conjecture. One has

$${\mathbf{E}}\lambda_i=n^{1/2}\gamma_i+n^{-1/2}C_{i,n}+\frac{1}{4\sqrt{n}}(\gamma_i^3-2\gamma_i){\mathbf{E}}\eta^4+O_\delta(n^{-1/2-c}).$$

The same asymptotic should hold with ${\mathbf{E}}\lambda_i$ replaced by the median ${\bf M}\lambda_i$.

The conjecture predicts that the fourth moment of the atom distribution contributes an explicit term to bulk eigenvalue locations, while the remaining bounded correction is independent of the atom distribution. The paper presents this as a heuristic refinement of the necessity of the four-moment hypothesis; its resolution status is not specified in the supplied source.