Universality for random tensors and cycle graphs with multiple edges

Abstract

We consider the universality for the trace invariants of $c_1 N \times \cdots \times c_D N$ tensors with i.i.d. complex random elements. In the case $c_1 = \cdots = c_D$, Gurau derived the universality in the limit $N \to \infty$ by representing the average trace invariants in terms of the corresponding colored graphs. Moreover he could explicitly calculate the asymptotic forms of the average trace invariants, if the corresponding graphs were special ones called "melonic graphs". In this paper we study another kind of special graphs, cycle graphs with multiple edges. One can construct these graphs by using simple cycle graphs and melonic graphs. Counting the number of the components, we can explicitly calculate the asymptotic forms of the average trace invariants.