Spiked Models in Wishart Ensemble [article]

Dong Wang
<span title="2008-04-06">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The spiked model is an important special case of the Wishart ensemble, and a natural generalization of the white Wishart ensemble. Mathematically, it can be defined on three kinds of variables: the real, the complex and the quaternion. For practical application, we are interested in the limiting distribution of the largest sample eigenvalue. We first give a new proof of the result of Baik, Ben Arous and Péché for the complex spiked model, based on the method of multiple orthogonal polynomials
more &raquo; ... Bleher and Kuijlaars. Then in the same spirit we present a new result of the rank 1 quaternionic spiked model, proven by combinatorial identities involving quaternionic Zonal polynomials (α = 1/2 Jack polynomials) and skew orthogonal polynomials. We find a phase transition phenomenon for the limiting distribution in the rank 1 quaternionic spiked model as the spiked population eigenvalue increases, and recognize the seemingly new limiting distribution on the critical point as the limiting distribution of the largest sample eigenvalue in the real white Wishart ensemble. Finally we give conjectures for higher rank quaternionic spiked model and the real spiked model.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0804.0889v1">arXiv:0804.0889v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mnwxuqll3zesfbf3iakrcb3pju">fatcat:mnwxuqll3zesfbf3iakrcb3pju</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-0804.0889/0804.0889.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0804.0889v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>