  We consider a Gaussian rotationally invariant ensemble of random real totally
symmetric tensors with independent normally distributed entries, and estimate
the largest eigenvalue of a typical tensor in this ensemble by examining the
rate of growth of a random initial vector under successive applications of a
nonlinear map defined by the random tensor. In the limit of a large number of
dimensions, we observe that a simple form of melonic dominance holds, and the
quantity we study is effectively determined by a single Feynman diagram arising
from the Gaussian average over the tensor components. This computation suggests
that the largest tensor eigenvalue in our ensemble in the limit of a large
number of dimensions is proportional to the square root of the number of
dimensions, as it is for random real symmetric matrices.
