Sharp estimates for large N Weingarten functions

Abstract

Weingarten functions provide a tool for computing Haar measure matrix integrals of polynomials in the matrix entries. An important property of Weingarten functions is their particularly simple large N limits. In 2017, Benoît Collins and Sho Matsumoto studied when this limit holds for Weingarten functions associated with integrals of products of 2n matrix entries, as n→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty $$\end{document}, together with the matrix size N [11]. They showed that the large N limit is uniformly achieved as long as n=o(N4/7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=o(N^{4/7})$$\end{document}, a result which already has applications to strong asymptotic freeness [2, 3]. However, their result is not optimal. They conjectured that their result should actually hold up to n=o(N2/3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=o(N^{2/3})$$\end{document} which is optimal. We prove this conjecture for the matrix groups G∈{U(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G \in \{\textrm{U}(N)$$\end{document}, O(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{O}(N)$$\end{document}, Sp(N)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Sp}(N)\}$$\end{document}. The proof proceeds by introducing a Markov process on permutations (pairings) which we call the unitary (orthogonal) Weingarten process. We believe this process may have further applications to the theory of Weingarten functions. We also prove two new bounds regarding the large N limit of the Weingarten function in the regimes when n=o(N4/5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=o(N^{4/5})$$\end{document}, and n=o(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=o(N)$$\end{document}.