Large deviations and stochastic calculus for large random matrices
Large deviations and stochastic calculus for large random matrices
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DOI：
10.1214/154957804100000033
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发表时间：
2004-09
影响因子：
1.6
通讯作者：
A. Guionnet
中科院分区：
文献类型：
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作者：
A. Guionnet
大型随机矩阵出现在数学和物理的不同领域，例如组合学、概率论、统计学、算子理论、数论、量子场论、弦理论等。在过去的十年中，它们引起了很多兴趣，特别是由于一系列数学突破，例如可以更好地理解其谱的局部特性、回答普适性问题、将这些问题与增长过程联系起来等。在本次调查中，我们将讨论这个问题 高斯随机矩阵的经验测量的大偏差，更一般地说，独立高斯随机矩阵的单词的踪迹。我们将描述这些问题是如何在物理学/组合学中通过研究所谓的矩阵模型或在自由概率中通过非交换熵的定义来激发的。我们将展示如何在这种情况下使用经典的大偏差技术。这些讲义应该可供非概率论者和非自由概率论者理解。
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the so-called matrix models or in free probability by the definition of a non-commutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non free-probabilists.