AI 中文总结
本文发展随机变量对的形状函数理论，证明双正则二分图上均匀支撑的随机变量对的形状函数与图关联矩阵算子范数的关系，推导其性质并实现数值近似。
AI 中文摘要
有限值随机变量对的形状函数由arXiv:2606.23849引入，该文献中用其推导了该对纠缠的谱界，纠缠是衡量可提取其互信息程度的量。本文进一步发展随机变量对的形状函数理论，证明当(X,Y)均匀支撑在双正则二分图的边上时，形状函数S(X,Y)(α,β)的值等于该图关联矩阵关于由(α,β)确定的勒贝格指数的算子范数的对数。该等价关系尤其可实现形状函数的数值近似，且通过对偶可得到该对的扩展轮廓（亦称张力区域）。本文还建立了形状函数满足的一系列关系与不等式，包括凸性与单调性性质、复合不等式，以及描述其在条件化和变量邻接下行为的关系。
英文摘要
The shape function of a pair of finite-valued random variables was introduced in arXiv:2606.23849, where it was used to derive a spectral bound on the entanglement of the pair, a quantity measuring the extent to which their mutual information can be extracted. In this article, we further develop the theory of shape functions for pairs of random variables. We prove that, when $(X,Y)$ is uniformly supported on the edges of a biregular bipartite graph, the value of the shape function $\mathcal{S}(X,Y)(α,β)$ equals the logarithm of the operator norm of the graph's incidence matrix with respect to Lebesgue exponents determined by $(α,β)$.This identification, in particular, enables the numerical approximation of the shape function and, by duality, of the extension profile, also known as the tension region, of the pair. We also establish a collection of relations and inequalities satisfied by shape functions, including convexity and monotonicity properties, composition inequalities, and relations describing their behavior under conditioning and the adjoining of variables.
