论文标题
大型随机张量的最大优势和最大的特征值
Melonic dominance and the largest eigenvalue of a large random tensor
论文作者
论文摘要
我们考虑具有独立正态分布的随机真实完全对称的张量的高斯旋转不变的集合，并通过检查由随机张紧器定义的非线性映射定义的非线性映射的连续应用中的随机初始载体在该集合中的典型张量最大的特征值。在大量维度的限制中，我们观察到一种简单的旋速优势形式存在，并且我们研究的数量是由张张量组件的高斯平均值产生的单个Feynman图有效确定的。该计算表明，我们的集合中最大的张量特征值在大量维度的极限上与尺寸数量的平方根成正比，因为它是随机的真实对称矩阵。
We consider a Gaussian rotationally invariant ensemble of random real totally symmetric tensors with independent normally distributed entries, and estimate the largest eigenvalue of a typical tensor in this ensemble by examining the rate of growth of a random initial vector under successive applications of a nonlinear map defined by the random tensor. In the limit of a large number of dimensions, we observe that a simple form of melonic dominance holds, and the quantity we study is effectively determined by a single Feynman diagram arising from the Gaussian average over the tensor components. This computation suggests that the largest tensor eigenvalue in our ensemble in the limit of a large number of dimensions is proportional to the square root of the number of dimensions, as it is for random real symmetric matrices.
