近日，日本冲绳科学技术大学院大学的Nicolas Delporte与京都大学的Naoki Sasakura合作并取得一项新进展。经过不懈努力，他们对具有偏差的随机张量特征值边缘进行研究。相关研究成果已于2025年1月14日在国际知名学术期刊《高能物理杂志》上发表。

对于大小为N的3阶对称高斯随机张量，在存在高斯噪声的情况下，研究人员延续文献[1]的工作，利用大N场论方法结合文献[2]早期的严格结果，计算了真实特征值分布和带符号特征值分布。研究人员描述了随着噪声方差的增加，两个分布边缘的行为特征。

研究人员发现了方差的两个临界值，第一个临界值对应于谱的主要部分中出现一个异常值，第二个临界值对应于该异常值与相应的最大特征值合并，并且它们都变得复杂。研究人员通过蒙特卡洛模拟支持了他们的结论。研究人员相信，这项研究结果为基于Z-特征值的随机张量伪谱的定义奠定了基础。

据悉，随机张量的最大特征值是包含无序系统的一个重要特征，相当于玻璃态系统的基态能量或量子态的注入范数。

附：英文原文

Title: The edge of random tensor eigenvalues with deviation

Author: Delporte, Nicolas, Sasakura, Naoki

Issue&Volume: 2025-01-14

Abstract: The largest eigenvalue of random tensors is an important feature of systems involving disorder, equivalent to the ground state energy of glassy systems or to the injective norm of quantum states. For symmetric Gaussian random tensors of order 3 and of size N, in the presence of a Gaussian noise, continuing the work [1], we compute the genuine and signed eigenvalue distributions, using field theoretic methods at large N combined with earlier rigorous results of [2]. We characterize the behaviour of the edge of the two distributions as the variance of the noise increases. We find two critical values of the variance, the first of which corresponding to the emergence of an outlier from the main part of the spectrum and the second where this outlier merges with the corresponding largest eigenvalue and they both become complex. We support our claims with Monte Carlo simulations. We believe that our results set the ground for a definition of pseudospectrum of random tensors based on Z-eigenvalues.

DOI: 10.1007/JHEP01(2025)071

Source: https://link.springer.com/article/10.1007/JHEP01(2025)071