In this note, we give a combinatorial and noncomputational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand–Tsetlin patterns with constraints.

中文翻译：

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随机酉矩阵的特征多项式的矩矩和格点数

在这篇笔记中，我们给出了哈尔分布酉矩阵的特征多项式矩的整数矩的渐近的组合和非计算证明，因为矩阵的大小趋于无穷大。这是通过将这些量与格点计数问题相关联来实现的。我们的主要结果是渐近中的前导阶系数的一个新的显式表达式，它是某个区域的体积，涉及具有约束的连续 Gelfand-Tsetlin 模式。