Asymptotic expansion of smooth functions in deterministic and iid Haar unitary matrices, and application to tensor products of matrices
Abstract
Let UN be a family of N×N independent Haar unitary random matrices and their adjoints, ZN a family of deterministic matrices, and P a self-adjoint noncommutative polynomial, that is, for any N, P(UN,ZN) is self-adjoint, f a smooth function. We prove that for any k, if f is smooth enough, there exist deterministic constants αiP(f,ZN) such that E[1 NTr(f(P(UN,ZN)))]=∑i=0kαiP(f,ZN)N2i+O(N−2k−2). Besides, the constants αiP(f,ZN) are built explicitly with the help of free probability. As a corollary, we prove that given α
Fields of Study & Subjects
Random Matrices and Applications
Advanced Algebra and Geometry
Spectral Theory in Mathematical Physics
Publication details
- Journal / source
- The Annals of Probability
- Publisher
- Institute of Mathematical Statistics
- Volume
- 54
- Issue
- 5
- Pages
- 2310–2376
- Publication date
- 28 August 2026
- Language
- EN