We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.

中文翻译：

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Wigner 矩阵多项式的局部定律

我们考虑几个独立的随机矩阵中的一般自伴随多项式，它们的条目居中并具有相同的方差。我们表明，在某些条件下，局部定律适用于最佳尺度，即特征值间距正上方尺度上的特征值密度遵循由自由概率论确定的全局状态密度。我们证明这些条件适用于一般二阶齐次多项式和具有 iid 项的独立矩阵的对称乘积，从而为这些类的系综建立了最优的局部定律。特别是，我们将 Anderson 的类似结果推广到反换向器。对于更一般的多项式，我们的条件在数值上是可有效检查的。