Spectral aspects of random heavy-tailed tensors

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 Erd\H{o}s-R\'enyi 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 Erd\H{o}s-R\'enyi $p$-uniform hypergraphs with edge probability $\alpha_N=c/N^{p-1}$. 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 $N^{p-1}\alpha_N\to\infty$ with $N^{p-1}(1-\alpha_N)\to\infty$, 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.