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

Maximilian Jerdee, Dmitriy Kunisky, Cristopher Moore

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.

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

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.
Paper Structure (5 sections, 4 theorems, 30 equations)

This paper contains 5 sections, 4 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Matchings and invariant polynomials
  4. Gurau's resolvent trace
  5. Construction of counterexample: Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

There exists $T \in \mathrm{Sym}^3(\mathbb{R}^{27})$ for which $\mathcal{I}_4(T) < 0$, and thus in particular for which there exists no probability measure $\gamma_T$ satisfying eq:muT.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: Matchings
  • Definition 2.2: Matching tensors
  • Definition 2.3: Invariant polynomials
  • Remark 2.4: Role in invariant theory
  • Proposition 2.5: Proposition 2.10 of bonnin2024universality
  • Definition 2.6
  • Proposition 2.7
  • proof
  • ...and 4 more