Usefulness of signed eigenvalue/vector distributions of random tensors

Abstract

Quantum field theories can be applied to compute various statistical properties of random tensors. In particular signed distributions of tensor eigenvalues/vectors are the easiest, which can be computed as partition functions of four-fermi theories. Though signed distributions are different from genuine ones because of extra signs of weights, they are expected to coincide in vicinities of ends of distributions. In this paper, we perform a case study of the signed eigenvalue/vector distribution of the real symmetric order-three random tensor. The correct critical point and the correct end in the large $N$ limit are obtained from the four-fermi theory, for which a method using the Schwinger-Dyson equation is very efficient. Since locations of ends are particularly important in applications, such as the largest eigenvalues and the best rank-one tensor approximations, signed distributions are the easiest and highly useful through the Schwinger-Dyson method.

Figures (3)

• Figure 1: Lefschets thimbles of (\ref{['eq:hc']}) for $x<x_c (|v|>|v|_c)$ (Left) and $x>x_c (|v|<|v|_c)$ (Right). Lefschetz thimbles ${\cal J}_\pm$ are shown by solid lines, while the upward flows ${\cal K}_\pm$ by dashed lines. The original contour ${\cal C}$ is crossed by both ${\cal K}_\pm$ for $|v|>|v|_c$ and only by ${\cal K}_-$ for $|v|<|v|_c$. This means that the two saddles $z_\pm$ contribute in the former case, but only $z_-$ for the latter.
• Figure 2: The size distribution of the smallest eigenvectors for $N=6,11$ (Left). The horizontal axis is rescaled as $\sqrt{N} v_{\rm smallest}$. Mean values of the size distributions $\sqrt{N} \langle v_{\rm smallest} \rangle$ are plotted against $N$ (Middle). The data can be fitted well with $\sqrt{N} \langle v_{\rm smallest} \rangle\sim 0.612\pm 0.011 + (0.00 \pm 0.10) \log N/N+ (0.69 \pm 0.11)/N$. The standard deviation $\sigma$ of the distribution can well be fitted with $\sqrt{N} \sigma \sim 0.63/N$ (Right).
• Figure 3: Left: The ratio $v_{\rm min}/\langle v_{\rm smallest}\rangle$ against $N$. It can be fitted with $1.000\pm 0.002-(0.23\pm 0.02)/N$.