论文标题
Harish-Chandra积分和$ S $转换的乘法对应物的渐近行为
Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform
论文作者
论文摘要
在本说明中，我们研究了球形积分的渐近线，这些渐近性是$β= 2 $的归一化schur多项式索引中的分析扩展，否则，杰克对称多项式。这样的积分是Harish-Chandra-Itzykson-Zuber（HCIZ）积分的乘法对应物，当一个矩阵之一是排名第一时，其渐近性是由所谓的$ r $ transform给出的。我们通过鞍点分析认为，在乘法案例中，所有$β> 0 $的类似结果均能成立，在乘法情况下，渐近造成的结果受$ s $转换的对数控制。由于这一结果，人们可以计算完全均匀对称多项式的渐近行为。
In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for $β=2$ , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called $R$-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all $β>0$ in the multiplicative case, where the asymptotic is governed by the logarithm of the $S$-transform. As a consequence of this result one can calculate the asymptotic behavior of complete homogeneous symmetric polynomials.
