Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory

Naoki Sasakura

doi:10.1016/j.physletb.2022.137618

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arXiv · 2208.08837

Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory

Naoki Sasakura

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Abstract

Eigenvalue distributions are important dynamical quantities in matrix models, and it is a challenging problem to derive them in tensor models. In this paper, we consider real symmetric order-three tensors with Gaussian distributions as the simplest case, and derive an explicit formula for signed distributions of real tensor eigenvectors: Each real tensor eigenvector contributes to the distribution by $\pm 1$, depending on the sign of the determinant of an associated Hessian matrix. The formula is expressed by the confluent hypergeometric function of the second kind, which is obtained by computing a partition function of a four-fermi theory. The formula can also serve as lower bounds of real eigenvector distributions (with no signs), and their tightness/looseness are discussed by comparing with Monte Carlo simulations. Large-$N$ limits are taken with the characteristic oscillatory behavior of the formula being preserved.

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Naoki Sasakura. 2022-08-18. Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory. https://doi.org/10.1016/j.physletb.2022.137618

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