Let $M$ be a matroid on a finite ground set $E$, and suppose that the automorphism group of $M$ acts transitively on $E$. We show the following: if $X_1,\ldots,X_K$ are sampled independently from a distribution $p$ on $E$, then the probability that the samples are distinct and that $\{X_1,\ldots,X_K\}$ is an independent set in $M$ is quasi-concave in $p$ and maximized when $p$ is uniform. As a corollary, for a random $K\times N$ matrix over a finite field whose rows are sampled independently from an arbitrary distribution on nonzero projective row classes, the uniform distribution on projective space maximizes the probability of full row rank. In this particular case we also establish the uniqueness of the maximizer and global quadratic stability, while a simple example illustrates that uniqueness and stability need not hold for arbitrary transitive matroids.

翻译：设$M$是有限基集$E$上的一个拟阵，并假设$M$的自同构群在$E$上传递作用。我们证明如下结论：若$X_1,\ldots,X_K$独立地取自$E$上的分布$p$，则样本互异且$\{X_1,\ldots,X_K\}$构成$M$中独立集的概率关于$p$是拟凹的，且在$p$为均匀分布时达到最大值。作为推论，对于有限域上随机$K\times N$矩阵（其行向量独立地取自非零射影行空间类上的任意分布），射影空间上的均匀分布使得矩阵行满秩的概率最大。针对该特例，我们还证明了最大化器的唯一性及全局二次稳定性，而一个简单示例表明，对于任意传递拟阵，唯一性与稳定性未必成立。