We show how the replica method can be used to compute the asymptotic eigenvalue spectrum of a real Wishart product matrix. This provides a compact, elementary derivation of a polynomial condition on the Stieltjes transform first proved by M\"uller [IEEE Trans. Inf. Theory. 48, 2086-2091 (2002)]. We additionally derive polynomial conditions on the minimum and maximum eigenvalues, which match the results obtained by Akemann, Ipsen, and Kieburg [Phys. Rev. E 88, 052118 (2013)] for the complex Wishart product ensemble.

中文翻译：

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真实Wishart乘积矩阵特征值的复制方法

我们展示了如何使用复制方法来计算真实 Wishart 乘积矩阵的渐近特征值谱。这提供了 M\"uller [IEEE Trans. Inf. Theory. 48, 2086-2091 (2002)] 首次证明的 Stieltjes 变换上多项式条件的紧凑、基本推导。我们还推导了关于最小值和最大值的多项式条件特征值，这与 Akemann、Ipsen 和 Kieburg [Phys. Rev. E 88, 052118 (2013)] 获得的复杂 Wishart 产品集合的结果相匹配。