持久性Lotka–Volterra群落中随机矩阵普适性的失效
Breakdown of random-matrix universality in persistent Lotka--Volterra communities
AI总结：
本文以广义Lotka–Volterra方程为背景，利用动态平均场理论证明生态群落中物种间相互作用呈现非高斯统计，导致随机矩阵普适性原理失效，并指出只有纳入这些非高斯统计才能正确预测群落稳定性。
摘要：
随机矩阵的本征值谱通常只依赖于其元素的一阶矩和二阶矩，而不依赖于抽取这些元素的具体分布。在应用中，这一普适性原理的有效性常常未经证明就被假定成立。在本文中，我们在广义Lotka–Volterra方程的背景下提供了一个相关的反例。利用动态平均场理论，我们推导了演化后生态群落中物种之间相互作用的统计性质。随后我们表明，要正确预测本征值谱并由此判断稳定性，需要这些相互作用的完整统计信息，而不仅仅是高斯系综所描述的那部分统计信息。因此，普适性原理在该系统中失效。我们由此证明，随机矩阵的本征值谱可用于推断“可行”生态群落的稳定性，但前提是必须考虑物种间相互作用所涌现出的非高斯统计特性。
英文摘要：
The eigenvalue spectrum of a random matrix often only depends on the first and second moments of its elements, but not on the specific distribution from which they are drawn. The validity of this universality principle is often assumed without proof in applications. In this letter, we offer a pertinent counterexample in the context of the generalised Lotka--Volterra equations. Using dynamic mean-field theory, we derive the statistics of the interactions between species in an evolved ecological community. We then show that the full statistics of these interactions, beyond those of a Gaussian ensemble, are required to correctly predict the eigenvalue spectrum and therefore stability. Consequently, the universality principle fails in this system. We thus show that the eigenvalue spectra of random matrices can be used to deduce the stability of `feasible' ecological communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account.
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