Two partition-function approaches to non-symmetric random tensor eigenvalues

At large N, random matrices with Gaussian distributed entries follow the Wigner semicircular law for the distribution of their eigenvalues. Random tensors are of interest in contexts of d > 2 dimensional quantum theories but do not enjoy simple generalisations of eigenvalues. Work has recently been done by Gurau to extend Wigner’s law to totally symmetric random tensors, with an approach inspired by the partition function of a Gaussian p-spin model. Starting from Gurau’s approach, I will motivate and introduce two new attempts to define and study eigenvalues of non-symmetric random tensors through partition functions, at finite and large N. One approach, based on a definition of a characteristic function, will be related to Gurau’s distribution. The other, based on a permuted definition of eigenvalues, will hint at a universality with differently-computed distributions for symmetric and complex random tensors.