Characteristic polynomials of tensors via Grassmann integrals and distributions of roots for random Gaussian tensors

TL;DR

The paper introduces a novel notion of tensor characteristic polynomials defined via a Grassmann partition function, applicable to antisymmetric tensors and extendable to broad permutation symmetries. By averaging over Gaussian tensors and applying a large-$N$ saddle-point analysis, the authors show that the polynomial is monic of degree $N$, its roots form a finite spectrum, and their density converges to a Fuss-Catalan distribution; zeros also relate to tensor eigenvalue analogs and connect to Gurau's generalized Wigner semicircle law through a simple variable transformation. A key result is the equivalence between the generating function of root powers and the partition function's derivative, linking spectral data to tensor invariants and to the combinatorics of Fuss-Catalan numbers. Overall, the work provides a new spectral framework for tensors that reveals universal random-matrix-like structures and suggests avenues for applying Grassmann methods to tensor decompositions and quantum-inspired tensor models.

Abstract

We propose a new definition of characteristic polynomials of tensors based on a partition function of Grassmann variables. This new notion of characteristic polynomial addresses general tensors including totally antisymmetric ones, but not totally symmetric ones. Drawing an analogy with matrix eigenvalues obtained from the roots of their characteristic polynomials, we study the roots of our tensor characteristic polynomial. Unlike standard definitions of eigenvalues of tensors of dimension $N$ giving $\sim e^{\text{constant} \, N}$ number of eigenvalues, our polynomial always has $N$ roots. For random Gaussian tensors, the density of roots follows a generalized Wigner semi-circle law based on the Fuss-Catalan distribution, introduced previously by Gurau [arXiv:2004.02660 [math-ph]].

Figures (4)

• Figure 2: We represent in the complex $q$-plane, the Lefschetz thimbles ending in the light blue regions and their duals ending in the brown regions, for each saddle point of the Fuss-Catalan equation, taking the action $S[q]=q-z q^p/p-\log (q/z^{1/p})$. The light blue (respectively brown) regions indicate where the real part of the action given in \ref{['eq:largeNactionQ']} is negative (respectively positive). The black points correspond to the $p$ saddle points given by the $p$ solutions of the Fuss-Catalan equation \ref{['eq:FC']}. The black circle around the origin is the original curve $\mathcal{C}$\ref{['eq:actionQ']}. Only for $z > z_c$, i.e.,(b) and (e), the two saddles of the right for each (yellow and green thimbles and dual thimbles) contribute at leading order in $N$. In the other cases $z < z_c$, i.e., (a) (c) and (d), the saddle point on yellow thimble and dual thimble, contributes at leading order. We have taken $z=z_0e^{i\theta_0}$, $\theta_0=0.02$, with $z_c=2^2/3^3\approx 0.15$ and $z_0=0.03,0.23$ for $p=3$, and $z_c = 3^3/4^4 \approx 0.10$ and $z_0=0.01, 0.06,0.16$ for $p=4$.
• Figure 3: The generalized Wigner law $\rho_\text{Gurau}(y)$ for $p=3$.
• Figure 4: The distribution of the absolute value $r$ of the roots from \ref{['eq:TensorWignerLaw']} superimposed with the histogram of the absolute value of the roots of the polynomial \ref{['eq:partitionfunctionOurs']}, for $N=2000$, $p=4$ and $\tilde{\mu}=1/p$.
• Figure : Plotting all the zeros of the partition function \ref{['eq:partitionfunctionmonicpolyn']} in the complex $\lambda$ plane, with $\mu=N^{1-p}/p$ (i.e., $\tilde{\mu} = 1/p$) for the values $p=2,3,4,5,6,7$ and $N=50$.