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PreprintarXiv licencearXiv

Matrix group Λ-distributions

  • Matthew Bertucci
  • Sam Van Blarcom
  • Benjamin Glancy
  • Sean Howe
  • Thomas Madden
  • Ben Miller
  • Souparna Pal
  • Giorgos Papagrigoriou
  • Nathan Raikman
  • Tien Tran
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Abstract

The Λ-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field L-functions. It is encoded by the σ-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the σ-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic Λ-distributions for the infinite families of Weyl reflection groups of types Bn/Cn and Dn, complementing the previously known case of reflection groups of type An, i.e., symmetric groups. We also establish a general result relating the shapes of σ-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.

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Cite

BibTeXRIS
BibTeX
@misc{bertucci2026matrix,
  title = {{Matrix group $Λ$-distributions}},
  author = {Matthew Bertucci and Sam Van Blarcom and Benjamin Glancy and Sean Howe and Thomas Madden and Ben Miller and Souparna Pal and Giorgos Papagrigoriou and Nathan Raikman and Tien Tran},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.16796},
  eprint = {2608.16796},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.16796},
}
RIS
TY  - UNPB
TI  - Matrix group Λ-distributions
AU  - Matthew Bertucci
AU  - Sam Van Blarcom
AU  - Benjamin Glancy
AU  - Sean Howe
AU  - Thomas Madden
AU  - Ben Miller
AU  - Souparna Pal
AU  - Giorgos Papagrigoriou
AU  - Nathan Raikman
AU  - Tien Tran
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2608.16796
UR  - https://arxiv.org/abs/2608.16796
AB  - The Λ-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field L-functions. It is encoded by the σ-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the σ-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic Λ-distributions for the infinite families of Weyl reflection groups of types B_n/C_n and D_n, complementing the previously known case of reflection groups of type A_n, i.e., symmetric groups. We also establish a general result relating the shapes of σ-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.
ER  -