An Extension of the HarishChandra-Itzykson-Zuber Integral
An Extension of the HarishChandra-Itzykson-Zuber Integral
复制标题
HarishChandra-Itzykson-Zuber 积分的扩展
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
S. Hikami
中科院分区：
文献类型：
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作者：
É. Brézin;S. Hikami
翻译后摘要：在酉群U（k）（β=2）的HarishChandra-Itzykson-Zuber积分存在于许多问题，涉及埃尔米特随机矩阵。 众所周知，该结果是半经典精确的。这个简单的结果不能推广到其他对称群，例如辛群或正交群。本文首先将这种积分的分析推广到辛群Sp（k）（β=4）上。在那里，半经典近似必须通过WKB展开来校正。事实证明，这种扩展在有限数量的项之后停止;换句话说，WKB近似由适当变量的多项式校正。分析是基于新的解决方案的热核微分方程。我们还研究了表征对称群的参数β的任意值。对于任意的β和k=3，以及对于大的β和任意的k，导出了封闭公式。
Abstract: The HarishChandra-Itzykson-Zuber integral over the unitary group U(k) (β=2) is present in numerous problems involving Hermitian random matrices. It is well known that the result is semi-classically exact. This simple result does not extend to other symmetry groups, such as the symplectic or orthogonal groups. In this article the analysis of this integral is extended first to the symplectic group Sp(k) (β=4). There the semi-classical approximation has to be corrected by a WKB expansion. It turns out that this expansion stops after a finite number of terms ; in other words the WKB approximation is corrected by a polynomial in the appropriate variables. The analysis is based upon new solutions to the heat kernel differential equation. We have also investigated arbitrary values of the parameter β, which characterizes the symmetry group. Closed formulae are derived for arbitrary β and k=3, and also for large β and arbitrary k.