A homological approach to the Gaussian Unitary Ensemble
Abstract
We study the Gaussian Unitary Ensemble (GUE) using noncommutative geometry and the homological framework of the BatalinVilkovisky (BV) formalism. Coefficients of the correlation functions in the GUE with respect to the rank $N$ are described in terms of ribbon graph Feynman diagrams that then lead to a counting problem for the corresponding surfaces. The canonical relations provided by this homological setup determine a recurrence relation for these correlation functions. Using this recurrence relation and properties of the Catalan numbers, we determine the leading order behavior of the correlation functions with respect to the rank $N$. As an application, we prove a generalization of Wigner's semicircle law and compute all the large $N$ statistical correlations for the family of random variables in the GUE defined by multitrace functions.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.04256
 Bibcode:
 2022arXiv220604256G
 Keywords:

 Mathematical Physics;
 Mathematics  Probability;
 Mathematics  Quantum Algebra;
 15B52;
 60B20;
 60F99;
 81T18;
 81T32;
 81T40;
 81T70;
 81T75
 EPrint:
 29 pages, 1 figure