论文标题
高斯张量模型的真实张量特征值/矢量分布
Real tensor eigenvalue/vector distributions of the Gaussian tensor model via a four-fermi theory
论文作者
论文摘要
特征值分布是矩阵模型中重要的动力量，在张量模型中研究相应数量是一个有趣的挑战。我们将实际张量特征值/矢量分布研究用于实际对称顺序的三个随机张量，并将高斯分布作为最简单的情况。我们首先将此问题重写为使用$ r $复制的费米子的四个fermi理论的分区函数的计算。分区功能是针对某些小$ n，r $案例的精确计算的，并且被证明与蒙特卡洛模拟完全一致。对于总$ n $，似乎很难准确地计算它，并且我们使用自搭配方程进行两点函数应用近似值并获得分析表达式。事实证明，通过取$ r = 1/2 $获得的实际张量特征值分布只是此近似值内的高斯。我们将近似表达式与蒙特卡洛模拟进行比较，发现，如果将额外的总体因素取决于$ n $与表达式相乘，它与蒙特卡洛的结果非常吻合。将来要进行研究，以改善大$ $ n $的近似值，以正确得出总体因素。
Eigenvalue distributions are important dynamical quantities in matrix models, and it is an interesting challenge to study corresponding quantities in tensor models. We study real tensor eigenvalue/vector distributions for real symmetric order-three random tensors with the Gaussian distribution as the simplest case. We first rewrite this problem as the computation of a partition function of a four-fermi theory with $R$ replicated fermions. The partition function is exactly computed for some small-$N,R$ cases, and is shown to precisely agree with Monte Carlo simulations. For large-$N$, it seems difficult to compute it exactly, and we apply an approximation using a self-consistency equation for two-point functions and obtain an analytic expression. It turns out that the real tensor eigenvalue distribution obtained by taking $R=1/2$ is simply the Gaussian within this approximation. We compare the approximate expression with Monte Carlo simulations, and find that, if an extra overall factor depending on $N$ is multiplied to the the expression, it agrees well with the Monte Carlo results. It is left for future study to improve the approximation for large-$N$ to correctly derive the overall factor.
