离散正交多项式系综与Plancherel测度
Discrete orthogonal polynomial ensembles and the Plancherel measure
摘要：
我们考虑离散正交多项式系综，它是随机矩阵理论中正交多项式系综的离散类似物。这些集合出现在组合概率的某些问题中，可以被认为是分区上的概率测度。Meixner系综与二维定向增长模型有关，Charlier系综与随机字中弱递增序列的长度有关。Krawtchouk系综与阿兹特克钻石的随机多米诺骨牌中的锯齿形路径有关，也与某种简化的定向第一通道渗透模型有关。本文利用Charlier系综研究了随机字中弱递增连续性的渐近性，并证明了Tracy和Widom的一个猜想。作为Meixner系综或Charlier系综的一个极限，我们得到了分区上的Plancherel测度，并利用这一点，我们证明了Baik，Deift和Johansson的一个猜想，即在Plancherel测度下，分区中前k行的长度分布，适当缩放，收敛于高斯酉系综中随机矩阵k个最大特征值的渐近联合分布。在这个问题中，一定的离散核，我们称之为离散贝塞尔核，起着重要的作用。
英文摘要：
We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a two-dimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zig-zag paths in random domino tilings of the Aztec diamond, and also in a certain simplified directed first-passage percolation model. We use the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of Tracy and Widom. As a limit of the Meixner ensemble or the Charlier ensemble we obtain the Plancherel measure on partitions, and using this we prove a conjecture of Baik, Deift and Johansson that under the Plancherel measure, the distribution of the lengths of the first k rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the Gaussian Unitary Ensemble. In this problem a certain discrete kernel, which we call the discrete Bessel kernel, plays an important role.
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