Asymptotic Properties of Random Contingency Tables with Uniform Margin

Let $C\geq 2$ be a positive integer. Consider the set of $n\times n$ non-negative integer matrices whose row sums and column are all equal to $Cn$ let $X=(X_{ij})_{1\leq i,j\leq n}$ uniformly distributed on this set. This $X$ is called random contingency table with uniform margin. In paper, we study various asymptotic properties as $n\to\infty$.