Spectral aspects of random heavy-tailed tensors

2026/07/27 by Remi Bonnin, Alexis Imbert
#math.PR

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Abstract

We investigate heavy-Wigner tensors: symmetric random tensors whose independent entries, up to the tensor symmetries, are centered and have moments of order N^-(p-1), where N is the tensor dimension. This framework includes normalized adjacency tensors of sparse Erdős-Rényi hypergraphs and truncated heavy-tailed tensor models. We study trace invariants, a complete family of polynomial invariants under permutations of the tensor indices. We prove that, after the natural normalization, the only non-vanishing asymptotic contributions are those associated with fat hypertrees, and we derive a central limit theorem for these injective trace invariants. As applications, we first analyze Erdős-Rényi p-uniform hypergraphs with edge probability α_N=c/N^p-1. We prove local weak convergence to a uniform Galton-Watson hypertree with Poisson offspring distribution. We also prove convergence of the empirical spectral distribution of the matrix obtained by contracting the adjacency tensor; in the sparse regime the limiting law depends on the sparsity parameter c and has unbounded support, while in the regime N^p-1α_N→∞ with N^p-1(1-α_N)→∞, the limiting spectral distribution is the semicircle law. This result generalizes for matrices obtained by contracting an arbitrary heavy-Wigner tensor and we derive an explicit formula for the moments of the limiting spectral measure.

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