On the ω-circular *-distribution for complex unimodular ω
Abstract
We consider a notion of ω-circular *-distribution μω, defined in terms of a parameter ω∈ ℂ with | ω|=1. This *-distribution appears as limit law in an algebraic central limit theorem (CLT) for ω-commuting unitaries, and interpolates between the Fermionic setting (at ω= −1) and the Bosonic one (at ω= 1). On a combinatorial level, μω is described in terms of pair-partitions of sets {1, ... ,2n}, where one has a total order among the n pairs of the partition and where every pair is assigned a direction, to the effect that one can define the sign of a crossing between two pairs. The main goal of the paper is to study a significant subspace, with "Fock space" flavour, of the L2-space associated to a natural multi-variate version of μω. The inner product on this subspace is defined by a sesquilinear form ⟨ · , · ⟩ω on non-commutative polynomials. We explicitly describe ⟨ · , · ⟩ω in terms of some tractable non-negative definite functions on symmetric groups, and we prove that it is non-degenerate if and only if ω is not a root of unity.
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BibTeX
@misc{koestler2026ωcircular,
title = {{On the $ω$-circular $*$-distribution for complex unimodular $ω$}},
author = {Claus Koestler and Liam McQuay and Alexandru Nica},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24443},
eprint = {2610.24443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24443},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On the ω-circular *-distribution for complex unimodular ω
AU - Claus Koestler
AU - Liam McQuay
AU - Alexandru Nica
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24443
UR - https://arxiv.org/abs/2610.24443
AB - We consider a notion of ω-circular *-distribution μ_ω, defined in terms of a parameter ω∈ ℂ with | ω|=1. This *-distribution appears as limit law in an algebraic central limit theorem (CLT) for ω-commuting unitaries, and interpolates between the Fermionic setting (at ω= −1) and the Bosonic one (at ω= 1). On a combinatorial level, μ_ω is described in terms of pair-partitions of sets {1, ... ,2n}, where one has a total order among the n pairs of the partition and where every pair is assigned a direction, to the effect that one can define the sign of a crossing between two pairs. The main goal of the paper is to study a significant subspace, with "Fock space" flavour, of the L^2-space associated to a natural multi-variate version of μ_ω. The inner product on this subspace is defined by a sesquilinear form ⟨ · , · ⟩_ω on non-commutative polynomials. We explicitly describe ⟨ · , · ⟩_ω in terms of some tractable non-negative definite functions on symmetric groups, and we prove that it is non-degenerate if and only if ω is not a root of unity.
ER -