Spectral aspects of random heavy-tailed tensors

We introduce heavy-Wigner random tensors and establish the convergence of a generating family of invariant linear forms in their entries together with a central limit theorem. The relevant group actions arise naturally in the study of tensor eigenvalues. Then, we compute the limiting spectral measure of the matrix obtained by contraction of a heavy-Wigner tensor and we derive two applications. The first one concerns the adjacency tensor of a Erd\H{o}s-R'enyi hypergraph. We identify the local weak limit of the hypergraph and study the limiting spectral distribution of the matrix contraction. The second application concerns L'evy tensors whose entries belong to the domain of attraction of an $\alpha$-stable law.