Freeness for tensors

TL;DR

This paper develops a free probability framework for tensors, extending classical concepts to high-dimensional random tensors and demonstrating their asymptotic freeness as dimensions grow.

Contribution

It introduces a novel notion of freeness for tensors of various orders and establishes its relevance in high-dimensional random tensor analysis.

Findings

Random tensor models are asymptotically free as dimension increases.

Defined free cumulants for tensors of different orders.

Established Schwinger-Dyson equations for random tensors.

Abstract

We pursue the current developments in random tensor theory by laying the foundations of a free probability theory for tensors and establish its relevance in the study of random tensors of high dimension. We give a definition of freeness associated to a collection of tensors of possibly different orders. Our definition reduces to the usual freeness when only tensors of order 2 are concerned. We define the free cumulants which are associated to this notion of tensor freeness. We prove that the basic models of random tensors are asymptotically free as the dimension goes to infinity. On the way, we establish Schwinger-Dyson loop equations associated to random tensors.