We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].

中文翻译：

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关于自由加性卷积的支持

我们考虑两个概率措施无添加剂卷积μ和ν对实体线显示，μ ⊞ v是否支持在单个区间μ和ν每个人都有一个区间的支持。此外，密度μ ⊞ ν被证明消失作为其支撑体的边缘附近平方根如果两个μ和ν有-1和1之间的边缘附近的指数幂律行为。特别是，这些结果表明，在我们最近的工作中，关于在频谱边缘添加随机矩阵的最佳局部定律的条件无处不在[5]。