Exact analytic expressions of real tensor eigenvalue distributions of Gaussian tensor model for small $N$

TL;DR

This paper derives exact formulas for the distributions of eigenvalues of Gaussian symmetric tensors for small sizes, using advanced integral techniques, and compares them with simulations to validate the results.

Contribution

It provides the first explicit analytic expressions for real tensor eigenvalue distributions of Gaussian tensors for small N, extending understanding in tensor spectral theory.

Findings

Exact distributions obtained for N ≤ 8

Good agreement with Monte Carlo simulations

Extrapolated large-N expressions proposed

Abstract

We obtain exact analytic expressions of real tensor eigenvalue/vector distributions of real symmetric order-three tensors with Gaussian distributions for $N\leq 8$. This is achieved by explicitly computing the partition function of a zero-dimensional boson-fermion system with four-interactions. The distributions are expressed by combinations of polynomial, exponential and error functions as results of feasible complicated bosonic integrals which appear after fermionic integrations. By extrapolating the expressions and also using a previous result, we guess a large-$N$ expression. The expressions are compared with Monte Carlo simulations, and precise and good agreement are obtained with the exact and the large-$N$ expressions, respectively. Understanding the feasibility of the integration is left for future study, which would provide a general-$N$ analytic formula.