In this talk we will develop a combinatorial approach for studying
moments of the resolvent trace for random tensors proposed by Razvan
Gurau, giving the maximal contribution of the expectation of trace
invariants (or moments) of a tensor. Our work is based on the study of
hypergraphs and extends the combinatorial proof of moments convergence
for Wigner's theorem. This also opens up paths for research akin to free
probability for random tensors.
Specifically, trace invariants form a complete basis of tensor
invariants and constitute the moments of the resolvent trace. For a
random tensor with entries independent, centered, with the right
variance and bounded moments, we will show the convergence of the
expectation and bound the variance of the balanced single trace
invariant. This implies the universality of the convergence of the
associated measure towards the law obtained by Gurau in the Gaussian
case, whose limiting moments are given by the Fuss-Catalan numbers.
Additionally, in the Gaussian case, the limiting distribution
of the $k$-times contracted $p$-order random tensor by a deterministic
vector is always the Wigner-Gurau law at order $p-k$, dilated by
$\sqrt{\binom{p-1}{k}}$.
