Intermediate Field Representation for Positive Matrix and Tensor Interactions

TL;DR

This paper introduces an intermediate field representation for positive random matrices and tensors with higher-degree interactions, enabling non-perturbative analysis and Borel-Le Roy summability proofs, advancing understanding of such models.

Contribution

It presents a novel intermediate field representation that respects positivity symmetry and proves Borel-Le Roy summability for these models, a significant step beyond previous methods.

Findings

Established a non-perturbative intermediate field representation.

Proved Borel-Le Roy summability of the models.

Identified limitations in associating a convergent Loop Vertex Expansion.

Abstract

In this paper we introduce an intermediate field representation for random matrices and random tensors with positive (stable) interactions of degree higher than 4. This representation respects the symmetry axis responsible for positivity. It is non-perturbative and allows to prove that such models are Borel-Le Roy summable of the appropriate order in their coupling constant. However we have not been able yet to associate a convergent Loop Vertex Expansion to this representation, hence our Borel summability result is not of the optimal expected form when the size N of the matrix or of the tensor tends to infinity.