Analytic aspects of free convolution

过去十年见证了无线通信理论及其在无线系统中的实现的许多进展。由于无线网络的时变特性，具有随机采样条目的矩阵（所谓的随机矩阵）自然会发挥作用。研究人员发现，许多多用户无线系统的性能测量与相应的随机矩阵模型的限制行为有关，因为矩阵的大小没有限制。 自由概率论为研究这种行为提供了一个框架。这个理论是由Dan Voiculescu在1985年为了回答纯数学问题而创立的。他后来发现，它也可以用来研究大型随机矩阵。这一非凡的发现极大地改变了自由概率，并为随机矩阵理论带来了新的工具和概念。此外，近年来工程师开始应用自由概率工具来解决无线通信中的问题，这使得自由概率成为应用科学家感兴趣的重要领域。 本文主要研究自由概率论的一个特殊分支--有限维代数上的自由度。本研究的动机是在无线通信中使用的一些随机矩阵模型。我计划用矩阵分析和自由熵的方法研究算子值半圆变量和自由卷积的正则性问题。我希望这方面的研究取得进一步的进展，将有利于理论和应用科学。

The last decade has witnessed many advances in wireless communication theory and their implementation in wireless systems. Matrices with randomly sampled entries, the so-called random matrices, come naturally into the play due to the time varying characteristics of the wireless network. Researchers found that the performance measurement for many multiuser wireless systems is relative to the limiting behavior of the corresponding random matrix models as the size of the matrices go without bound. Free probability theory provides a framework for studying such a behavior. This theory was created by Dan Voiculescu in 1985 in order to answer problems in pure mathematics. He later found that it can also be used to study large random matrices. This remarkable discovery changed free probability dramatically and brought new tools and concepts into the random matrix theory. Moreover, in recent years engineers started to apply free probability tools to solve problems in wireless communications, and this has made free probability an interesting and important field for applied scientists. This proposal is mainly concerned with research projects in a special branch of free probability theory, the freeness with amagalmation over a finite dimensional algebra. The motivation of this research is some random matrix models used in wireless communications. I plan to investigate the regularity issues for the operator-valued semicircular variables and for free convolution by methods in matrix analysis and free entropy. I hope that further progress in this line of research would benefit both theoretical and applied sciences.