Characterization of Gaussian Tensor Ensembles

2025/05/05 by Bonnin, Remi
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2505.02814

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 basis 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.

Citations

• Tensorial free convolution, semicircular, free Poisson and R-transform in high order
• Tensor cumulants for statistical inference on invariant distributions
• Universality of the Wigner-Gurau limit for random tensors
• Signed eigenvalue/vector distribution of complex order-three random tensor
• Spherical Matrix Ensembles
• Universality for Random Tensors

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