随机矩阵渐近自由独立性问题已成为非交换概率理论的研究热点与前沿。到目前为止，对随机矩阵渐近自由独立性的研究主要是基于随机矩阵的元素为可交换随机变量，而关于非交换随机变量组成的随机矩阵渐近自由独立性研究尚处于起步阶段，亟需对其进一步的探索与研究。我们将运用自由累积、非交错分拆、Cauchy-Stieltjes变换等工具深入研究非交换随机变量构成的随机矩阵渐近自由独立性问题。更确切地说，本项目将研究自由独立同分布R-对角非交换随机变量组成的随机矩阵渐进自由独立性、二阶及高阶渐近自由独立性；不全为自由独立同分布的非交换随机变量组成的随机矩阵渐近自由独立性、二阶及高阶渐近自由独立性；满足鞅性质的非交换随机变量组成的随机矩阵渐近自由独立性、二阶及高阶渐近自由独立性。该项目所得相关结论对二阶自由概率理论的建立将起着重要推动作用，也对非交换概率与随机矩阵理论的发展具有重要意义。

The asymptotic free independence property of random matrices is a research focus in Non-commutative probability theory. However, as so far, most works in this field are based on random matrix with entries in a commutative algebra. The study on asymptotic behavior of random matrices with non-commutative entries is still in the initial stage, and it needs to be further studied. This project will pay close attention to the deep study of some interesting results for the case when the entries are not commuting via free cumulant, non-crossing partition and Cauchy-Stieltjes transform techniques. More precisely, we will study asymptotic free independence, second order asymptotic freeness and higher order asymptotic freeness of random matrices with identically distributed and free independent R-diagonal (respectively, non-commutative dependent, non-commutative martingale structure) entries. The main results of the project will play an important role in developing the second order free probability theory, and promoting the theory of non-commutative probability and random matrix.