Orthogonal polynomial ensembles in probability theory
Orthogonal polynomial ensembles in probability theory
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DOI：
10.1214/154957805100000177
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发表时间：
2004-03
影响因子：
1.6
通讯作者：
W. Koenig
中科院分区：
文献类型：
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作者：
W. Koenig
我们调查了一些模型，从物理学，统计力学，概率论和组合学，这是每个描述的{它正交多项式系综}。最突出的例子显然是Hermite系综，高斯酉系综（GUE）的特征值分布，以及随机矩阵理论中其他众所周知的系综，如Wishart矩阵谱的Laguerre系综。近年来，人们发现了一些进一步有趣的模型，导致正交多项式系综，其中包括角增长模型，定向最后一次通过渗流，PNG液滴，非碰撞随机过程，随机排列的最长递增子序列的长度等。在大粒子数的限制下，这些模型的渐近行为的普适类，特别是粒子间的间距和最大粒子的涨落行为，受到了广泛的关注。计算机模拟表明，连接甚至更远，也包括黎曼zeta函数的零点。现有的证明需要大量的技术机器和重型工具，从各个部分的数学，特别是复杂的分析，组合和变分分析。特别是在过去十年中，取得了一些很好的成果，但显然对这一问题仍然缺乏全面和深入的理解。因此，现在似乎是一个适当的时间来提供一个调查的文本在这一研究领域。在本文中，我们介绍了各种模型，解释了问题和问题，并指出了模型之间的关系。此外，我们简明扼要地概述了一些最重要的结果的证明的一些要素。本文是针对非专家与强大的背景概率谁想要实现一个快速的调查领域。
We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an {it orthogonal polynomial ensemble}. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther and also comprise the zeros of the Riemann zeta function. The existing proofs require a substantial technical machinery and heavy tools from various parts of mathematics, in particular complex analysis, combinatorics and variational analysis. Particularly in the last decade, a number of fine results have been achieved, but it is obvious that a comprehensive and thorough understanding of the matter is still lacking. Hence, it seems an appropriate time to provide a surveying text on this research area. In the present text, we introduce various models, explain the questions and problems, and point out the relations between the models. Furthermore, we concisely outline some elements of the proofs of some of the most important results. This text is aimed at non-experts with strong background in probability who want to achieve a quick survey over the field.