We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization, Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the \textit{typical table}. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two dual variables, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in \( L^{1} \) to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$. We derive several new results for random contingency tables from our general framework.

翻译：我们研究具有独立同分布条目的随机大矩阵，这些矩阵被约束为具有指定的行和与列和（边缘）。该问题与相对熵最小化、薛定谔桥、列联表枚举以及具有给定度序列的随机图有着丰富的联系。我们证明，这种边缘约束的随机矩阵会急剧集中在一个特定的确定性矩阵周围，我们称之为 \textit{典型表}。典型表具有双重表征：(1) 在期望达到目标边缘的约束下，与基础模型具有最小相对熵的随机矩阵系综的期望；(2) 通过对基础模型进行秩一指数倾斜得到的最大似然模型的期望。典型表的结构由两个对偶变量决定，它们给出了倾斜参数的最大似然估计。基于这些结果，对于一系列"温和"的边缘，当矩阵尺寸趋于无穷大时，这些边缘在 \( L^{1} \) 范数下收敛于一个极限连续边缘，我们证明了边缘约束随机矩阵序列在割范数下收敛于一个极限核，该核是相应缩放典型表的 $L^{2}$-极限。收敛速度由边缘在 $L^{1}$ 范数下的收敛速度控制。我们从一般框架中推导出了关于随机列联表的若干新结果。