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Weingarten calculus for centered random permutation matrices
Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 465-479

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.480
Classification: 28C10, 15B52
Keywords: Weingarten calculus, Random permutation matrices
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Collins, Benoît; Nagatsu, Manasa. Weingarten calculus for centered random permutation matrices. Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 465-479. doi: 10.5802/alco.480
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[1] Handbook of mathematical functions, with formulas, graphs, and mathematical tables (Abramowitz, Milton; Stegun, Irene A., eds.), Dover Publications, Inc., New York, 1966, xiv+1046 pages | MR

[2] Banica, Teodor; Collins, Benoît Integration over compact quantum groups, Publ. Res. Inst. Math. Sci., Volume 43 (2007) no. 2, pp. 277-302 | MR | DOI | Zbl

[3] Banica, Teodor; Curran, Stephen; Speicher, Roland De Finetti theorems for easy quantum groups, Ann. Probab., Volume 40 (2012) no. 1, pp. 401-435 | DOI | MR | Zbl

[4] Bordenave, Charles A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts, Ann. Sci. Éc. Norm. Supér. (4), Volume 53 (2020) no. 6, pp. 1393-1439 | DOI | MR | Zbl

[5] Bordenave, Charles; Collins, Benoît Eigenvalues of random lifts and polynomials of random permutation matrices, Ann. of Math. (2), Volume 190 (2019) no. 3, pp. 811-875 | DOI | MR | Zbl

[6] Brannan, Michael; Collins, Benoît Dual bases in Temperley-Lieb algebras, quantum groups, and a question of Jones, Quantum Topol., Volume 9 (2018) no. 4, pp. 715-748 | DOI | MR | Zbl

[7] Collins, Benoît Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not. (2003) no. 17, pp. 953-982 | DOI | MR | Zbl

[8] Collins, Benoît; Matsumoto, Sho Weingarten calculus via orthogonality relations: new applications, ALEA Lat. Am. J. Probab. Math. Stat., Volume 14 (2017) no. 1, pp. 631-656 | DOI | MR | Zbl

[9] Collins, Benoît; Matsumoto, Sho; Novak, Jonathan The Weingarten calculus, Notices Amer. Math. Soc., Volume 69 (2022) no. 5, pp. 734-745 | DOI | MR | Zbl

[10] Collins, Benoît; Śniady, Piotr Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys., Volume 264 (2006) no. 3, pp. 773-795 | DOI | MR | Zbl

[11] Friedman, Joel A proof of Alon’s second eigenvalue conjecture and related problems, Mem. Amer. Math. Soc., Volume 195 (2008) no. 910, p. viii+100 | DOI | MR | Zbl

[12] Matsumoto, Sho; Novak, Jonathan Jucys-Murphy elements and unitary matrix integrals, Int. Math. Res. Not. IMRN (2013) no. 2, pp. 362-397 | DOI | MR | Zbl

[13] Stanley, Richard P. Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 1997, xii+325 pages | DOI | MR | Zbl

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