SIGMA 22 (2026), 031, 19 pages      arXiv:2505.02814      https://doi.org/10.3842/SIGMA.2026.031

Characterization of Gaussian Tensor Ensembles

Rémi Bonnin
Aix-Marseille Université, CNRS, I2M, Marseille, France

Received July 09, 2025, in final form March 13, 2026; Published online March 31, 2026

Abstract
The starting point of this work is a theorem due to Maxwell characterizing the distribution of a Gaussian vector with at least two coordinates. We define the Gaussian orthogonal, unitary and symplectic tensor ensembles for notions of real symmetric, hermitian and self-dual hermitian tensors which recover the classical vector and matrix Gaussian ensembles when the order is one and two. We give a complete family of invariant polynomials for orthogonal, unitary and symplectic transformations and we prove a Maxwell-type theorem for these Gaussian tensor distributions unifying and extending the ones known for vectors and matrices.

Key words: random tensors; Gaussian ensembles; Maxwell's theorem; orthogonal/unitary/symplectic invariance.

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