Random Discriminants
随机判别式
Let $X_1, X_2, \cdots, X_n$ be a random sample from a continuous univariate distribution $F$, and let $\Delta = \prod_{1 \leq i < j \leq n}(X_i - X_j)^2$ denote the discriminant, or square of the Vandermonde determinant, constructed from the random sample. The statistic $\Delta$ arises in the study of moment matrices and inference for mixture distributions, the spectral theory of random matrices, control theory and statistical physics. In this paper, we study the probability…