A homological approach to the Gaussian unitary ensemble

Abstract

We study the Gaussian unitary ensemble (GUE) using noncommutative geometry and the Batalin–Vilkovisky (BV) formalism, and we show how this homological approach provides canonical relations between correlation functions in the GUE. As applications of this method, we obtain new ways to prove generalizations of Wigner’s semicircle law, to compute all the large $N$ statistical correlations for multi-trace functions as random variables in the GUE, and to determine the leading (and subleading) order behavior of the correlation functions with respect to the rank $N$. Along the way, and illuminating the connection with prior work, we develop an explicit dictionary between this homological approach and the well-known combinatorics of ribbon graphs that lead to counting problems for the corresponding surfaces.