Weingarten calculus for centered random permutation matrices
Benoît Collins, Manasa Nagatsu
TL;DR
This work develops a Weingarten calculus tailored to centered random permutation matrices in $S_N$, revealing that a core building block is the quantity $a_k(N)$ connected to Kummer's confluent hypergeometric function. It provides a centered Weingarten formula with explicit combinatorial structure, establishes sign properties, and derives precise asymptotics that enhance understanding of moment decay and strong convergence phenomena. The paper also introduces recursive reduction techniques to simplify integrals to cases with $\Pi_{\mathbf{i}} \wedge \Pi_{\mathbf{j}}=0_k$, and constructs nontrivial uniform and nonuniform bounds that illuminate limitations in moment control for centered permutations. Collectively, these results deepen the algebraic and asymptotic analysis of moments of random permutations and clarify their role in related probabilistic and graph-theoretic contexts.
Abstract
We introduce and study the Weingarten calculus for centered random permutation matrices in the symmetric group S_N. After presenting a formulation of the Weingarten calculus on the symmetric group, we derive a formula in the centered case, as well as a sign-respecting formula. Our investigations uncover the fact that a building block of this Weingarten calculus is Kummer's confluent hypergeometric function. It allows us to derive multiple algebraic properties of the Weingarten function and uniform estimate. These results shed a conceptual light on phenomena that take place regarding the algebraic and asymptotic behavior of moments of random permutations in the resolution of Bordenave and Bordenave-Collins of strong convergence. We obtain multiple new non-trivial estimates for moments of coefficients in centered moments.