# Asymptotic resolvents of a product of two marginals of a random tensor

## TL;DR

This paper investigates the asymptotic behavior of the resolvent of a matrix formed from two marginals of a random tensor, revealing explicit algebraic equations and interpolating probability densities in different regimes.

## Contribution

It introduces a novel analysis of the resolvent for such matrices, deriving explicit algebraic equations and describing the density's interpolation between known laws.

## Key findings

- Resolvent computed explicitly in the first regime using free harmonic analysis.
- Algebraic degree six equation satisfied by the resolvent in the second regime.
- Density function interpolates between squared Marchenko-Pastur and its free multiplicative square.

## Abstract

Random tensors can be used to produce random matrices. This idea is, for instance, very natural when one studies random quantum states with the aim of exploring properties that are generically true, or true with some probability. We hereby study the moments generating function, in the sense of the Stieltjes transform - i.e. the resolvent -, of a random matrix defined as a product of two different marginals of the same random tensor. We study the resolvent in two different asymptotical regimes. In the first regime, the resolvent is easily computed thanks to freeness results for the two different marginals and straightforward application of free harmonic analysis. In the second regime, we show that the resolvent satisfies an algebraic equation of degree six. This algebraic equation possesses roots whose expressions can be given explicitly in terms of radicals. We obtain this result by using an enumerative combinatorics approach. One of the interesting aspects of the second regime is that the corresponding probability density function interpolates between the square of a Marchenko-Pastur and the free multiplicative square of a Marchenko-Pastur law.