Sharp estimates for large N Weingarten functions

TL;DR

This work resolves the uniform large-N regime for Weingarten functions associated with Haar integrals on the unitary, orthogonal, and symplectic groups by establishing a sharp bound that holds when n = o(N^{2/3}). It introduces the unitary (and orthogonal) Weingarten processes, a Markovian path-sampling framework on permutation (and pairing) graphs, and leverages this to control path counts via concentration-dominated cycle dynamics and Catalan-number relations. The main contribution is a tight bound of the form N^{n+| au|} Wg_N^G(τ)/Moeb(τ) ≤ 1/(1 − C n^p/N^q) with explicit constants, demonstrating optimality in the full-cycle case and providing nonuniform refinements for broader regimes (n = o(N^{4/5}) and n = o(N)). These results advance precise large-N asymptotics for Weingarten functions and have implications for strong asymptotic freeness and related matrix-model limits. The techniques yield a versatile probabilistic approach to Weingarten theory, potentially enabling further applications of the Weingarten process in random matrix theory and quantum information.

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 Benoit Collins and Sho Matsumoto studied when this limit holds for Weingarten functions associated to integrals of products of $2n$ matrix entries, as $n \to \infty$, together with the matrix size $N$. They showed that the large $N$ limit is uniformly achieved as long as $n=o(N^{4/7})$, a result which already has applications to strong asymptotic freeness. However, their result is not optimal. They conjectured that their result should actually hold up to $n=o(N^{2/3})$ which is optimal. We prove this conjecture for the matrix groups $G \in \{\mathrm{U}(N)$, $\mathrm{O}(N)$, $\mathrm{Sp}(N)\}$. The proof proceeds by introducing a Markov process on permutations (pairings) which we call the unitary (orthogonal) $\textit{Weingarten process}$. 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 $n=o(N^{4/5})$, and $n=o(N)$.

Figures (3)

• Figure 1: The figure above displays the unitary Weingarten graph $\mathcal{W}^{\mathrm{U}}$ restricted to layers 0 through 3, $\bigsqcup_{n=0}^{3} S_n$.
• Figure 2: The figure above displays the orthogonal Weingarten graph $\mathcal{W}^{\mathrm{O}}$ restricted to layers 0 through 2, $\bigsqcup_{n=0}^{2} \mathcal{P}_{2}(2n)$.
• Figure 3: Take $n = 8$, and consider $\pi = (1 \, 2)(3 \, 7)(4 \, 6)(5 \, 8)$, and $\tau = (4 \, 7)$ so that $\tau.\pi=(1 \, 2)(3 \, 4)(6 \, 7)(5 \, 8)$. Then $\mathcal{G}_{\pi}$ is depicted on the left with a Young diagram of the coset type of $\pi$ below. Similarly, $\mathcal{G}_{\tau.\pi}$ is depicted on the right with the Young Diagram of the coset type of $\tau.\pi$ below.

Theorems & Definitions (76)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Definition 2.1
• Definition 2.2
• Lemma 2.3