Universality for random tensors and cycle graphs with multiple edges

TL;DR

This paper extends the universality results for random tensors by analyzing cycle graphs with multiple edges, providing explicit asymptotic calculations for trace invariants beyond melonic graphs.

Contribution

It introduces a new class of graphs, cycle graphs with multiple edges, and derives explicit asymptotic formulas for their trace invariants, expanding the scope of universality in random tensor models.

Findings

Explicit asymptotic forms for trace invariants of cycle graphs with multiple edges.

Extension of universality results to new graph classes in tensor models.

Method to count components and compute invariants for complex graph structures.

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.