Gurau's spectral density is not a probability measure for individual real symmetric tensors

TL;DR

The paper demonstrates that Gurau's spectral density for individual real symmetric tensors does not correspond to a probability measure, contrasting with its probabilistic interpretation in random tensor ensembles.

Contribution

It shows that Gurau's spectral density, while meaningful on average for random tensors, is not a probability measure for specific deterministic tensors, highlighting limitations of the spectral density concept.

Findings

Gurau's spectral density aligns with probability measures on average for random tensors.

Constructed deterministic tensors where Gurau's spectral density is not a probability measure.

Highlights the difference between average-case and pointwise spectral properties.

Abstract

Gurau (2020) proposed a generalization of the trace of the matrix resolvent to tensors of higher order, and recent work has explored analogs of the Wigner semicircle and Marchenko-Pastur distributions from random matrix theory as well as aspects of free probability theory from this perspective. In particular, when evaluated with appropriate large random tensors, the limiting expectations of the coefficients of a series expansion of Gurau's resolvent trace give the moment sequences of probability measures analogous to the above distributions. We construct, on the other hand, individual deterministic tensors such that the same coefficients evaluated on those tensors do not give the moment sequence of any probability measure. Thus, the "spectral density" associated to Gurau's resolvent trace, while in a sense defined on average for certain random tensor ensembles, is not defined pointwise (unless perhaps as a signed measure) for all individual tensors.