Melonic dominance and the largest eigenvalue of a large random tensor

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Abstract

We consider a Gaussian rotationally invariant ensemble of random real totally symmetric tensors with independent normally distributed entries, and estimate the largest eigenvalue of a typical tensor in this ensemble by examining the rate of growth of a random initial vector under successive applications of a nonlinear map defined by the random tensor. In the limit of a large number of dimensions, we observe that a simple form of melonic dominance holds, and the quantity we study is effectively determined by a single Feynman diagram arising from the Gaussian average over the tensor components. This computation suggests that the largest tensor eigenvalue in our ensemble in the limit of a large number of dimensions is proportional to the square root of the number of dimensions, as it is for random real symmetric matrices.
Original languageEnglish
Article number66
Number of pages18
JournalLetters in Mathematical Physics
Volume111
Issue number3
DOIs
Publication statusPublished - Jun 2021

Bibliographical note

v2: comments and references added, accepted for publication

Keywords

  • math-ph
  • hep-th
  • math.MP
  • math.PR
  • math.SP

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