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PreprintCC BY 4.0arXiv

Loop Equations Characterize Random Matrix Statistics

  • Paul Bourgade
  • Jiaoyang Huang
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Abstract

We prove that the universal local point processes of random matrix theory are characterized by their loop equation hierarchies. More precisely, for every rational β>0, the Sineβ point process is the unique solution of the bulk loop equation hierarchy, and the Airyβ point process is the unique solution of the edge loop equation hierarchy. These uniqueness results provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble. In many models, these equations follow from local laws and integration by parts.

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Cite

BibTeXRIS
BibTeX
@misc{bourgade2026loop,
  title = {{Loop Equations Characterize Random Matrix Statistics}},
  author = {Paul Bourgade and Jiaoyang Huang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.07617},
  eprint = {2607.07617},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.07617},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Loop Equations Characterize Random Matrix Statistics
AU  - Paul Bourgade
AU  - Jiaoyang Huang
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2607.07617
UR  - https://arxiv.org/abs/2607.07617
AB  - We prove that the universal local point processes of random matrix theory are characterized by their loop equation hierarchies. More precisely, for every rational β>0, the Sine_β point process is the unique solution of the bulk loop equation hierarchy, and the Airy_β point process is the unique solution of the edge loop equation hierarchy. These uniqueness results provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble. In many models, these equations follow from local laws and integration by parts.
ER  -