Year: 2026
Type: preprint
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 $\mathrm{Sine}_β$ point process is the unique solution of the bulk loop equation hierarchy, and the $\mathrm{Airy}_β$ 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.
Source: arXiv (Cornell University)
Authors: Bourgade, Paul, Huang, Jiaoyang
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167841576