Complexification of $ z $-semicircular systems and orientation-averaged $ z $-circular laws

Ayman Alahmade; Ayman Alahmade

doi:10.3934/math.20261184

AIMS Mathematics

2026, Volume 11, Issue 9: 29848-29874. doi: 10.3934/math.20261184

Previous Article Next Article

Research article

Complexification of $ z $-semicircular systems and orientation-averaged $ z $-circular laws

Ayman Alahmade ^,

Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia

Received: 12 April 2026 Revised: 02 August 2026 Accepted: 25 August 2026 Published: 15 September 2026
MSC : 46L54, 46L53, 60B20

In this work, we study whether taking the complexification of a $ z $-semicircular system naturally gives rise to a $ z $-circular system. Starting from a semicircular family $ (s_r) $, we form complex variables $ c_r = \frac{1}{\sqrt{2}}\, (s_{2r-1} + i\, s_{2r}) $ and analyze their joint $ * $-moments using the combinatorics of pairings with oriented crossings. In the classical case where the deformation parameter is real $ q \in (-1, 1) $, this construction yields a $ q $-circular system, recovering the known correspondence between semicircular and circular variables. However, we show that this correspondence fails for general complex parameters $ z $ when $ |z| < 1 $. The obstruction is the additional averaging over balanced orientations, which agrees with the standard $ z $-circular law only when $ z $ is real. We show that the complexification of a $ z $-semicircular system introduces a new class of distributions, which we call orientation-averaged $ z $-circular systems, defined by averaging the contributions of oriented crossings over all admissible orientations. When $ z \in \mathbb{R} $, these systems coincide with the classical $ z $-circular systems. However, they are fundamentally different in the non-real case. As a result, we get a rigidity finding indicating that the only $ z $-circular systems formed by complexifying semicircular systems have a real deformation parameter.
- $ z $-circular systems,
- $ z $-semicircular systems,
- free probability,
- operator theory,
- noncommutative probability
Citation: Ayman Alahmade. Complexification of $ z $-semicircular systems and orientation-averaged $ z $-circular laws[J]. AIMS Mathematics, 2026, 11(9): 29848-29874. doi: 10.3934/math.20261184

Related Papers:

Abstract

In this work, we study whether taking the complexification of a $ z $-semicircular system naturally gives rise to a $ z $-circular system. Starting from a semicircular family $ (s_r) $, we form complex variables $ c_r = \frac{1}{\sqrt{2}}\, (s_{2r-1} + i\, s_{2r}) $ and analyze their joint $ * $-moments using the combinatorics of pairings with oriented crossings. In the classical case where the deformation parameter is real $ q \in (-1, 1) $, this construction yields a $ q $-circular system, recovering the known correspondence between semicircular and circular variables. However, we show that this correspondence fails for general complex parameters $ z $ when $ |z| < 1 $. The obstruction is the additional averaging over balanced orientations, which agrees with the standard $ z $-circular law only when $ z $ is real. We show that the complexification of a $ z $-semicircular system introduces a new class of distributions, which we call orientation-averaged $ z $-circular systems, defined by averaging the contributions of oriented crossings over all admissible orientations. When $ z \in \mathbb{R} $, these systems coincide with the classical $ z $-circular systems. However, they are fundamentally different in the non-real case. As a result, we get a rigidity finding indicating that the only $ z $-circular systems formed by complexifying semicircular systems have a real deformation parameter.

References

[1]	D. Voiculescu, Limit laws for random matrices and free products, Invent. Math., 104 (1991), 201–220. https://doi.org/10.1007/BF01245072 doi: 10.1007/BF01245072
[2]	D. V. Voiculescu, K. J. Dykema, A. Nica, Free random variables, Providence: American Mathematical Society, 1992. https://doi.org/10.1090/crmm/001
[3]	M. Bożejko, R. Speicher, An example of a generalized Brownian motion, Comm. Math. Phys., 137 (1991), 519–531. https://doi.org/10.1007/BF02100275 doi: 10.1007/BF02100275
[4]	M. Bożejko, B. Kümmerer, R. Speicher, $q$-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys., 185 (1997), 129–154. https://doi.org/10.1007/s002200050084 doi: 10.1007/s002200050084
[5]	J. A. Mingo, A. Nica, Random unitaries in non-commutative tori, and an asymptotic model for $q$-circular systems, Indiana Univ. Math. J., 50 (2001), 953–987.
[6]	R. Speicher, A non-commutative central limit theorem, Math. Z., 209 (1992), 55–66. https://doi.org/10.1007/BF02570820 doi: 10.1007/BF02570820
[7]	K. Dykema, A. Nica, On the Fock representation of the $q$-commutation relations, J. Reine Angew. Math., 440 (1993), 201–212. https://doi.org/10.1515/crll.1993.440.201 doi: 10.1515/crll.1993.440.201
[8]	M. Bożejko, R. Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann., 300 (1994), 97–120. https://doi.org/10.1007/BF01450478 doi: 10.1007/BF01450478
[9]	E. G. Effros, M. Popa, Feynman diagrams and Wick products associated with $q$-Fock space, Proc. Natl. Acad. Sci. USA, 100 (2003), 8629–8633. https://doi.org/10.1073/pnas.1531460100 doi: 10.1073/pnas.1531460100
[10]	P. Śniady, Gaussian random matrix models for $q$-deformed Gaussian variables, Comm. Math. Phys., 216 (2001), 515–537. https://doi.org/10.1007/s002200000345 doi: 10.1007/s002200000345
[11]	P. Śniady, Limit distributions of many-particle spectra and $q$-deformed Gaussian variables, Banach Center Publ., 72 (2006), 409–414.
[12]	A. Nou, Asymptotic matricial models and QWEP property for $q$-Araki-Woods algebras, J. Funct. Anal., 232 (2006), 295–327. https://doi.org/10.1016/j.jfa.2005.05.001 doi: 10.1016/j.jfa.2005.05.001
[13]	A. Nou, Non injectivity of the $q$-deformed von Neumann algebra, Math. Ann., 330 (2004), 17–38. https://doi.org/10.1007/s00208-004-0523-4 doi: 10.1007/s00208-004-0523-4
[14]	E. Ricard, Factoriality of $q$-Gaussian von Neumann algebras, Comm. Math. Phys., 257 (2005), 659–665. https://doi.org/10.1007/s00220-004-1266-5 doi: 10.1007/s00220-004-1266-5
[15]	M. Kennedy, A. Nica, Exactness of the Fock space representation of the $q$-commutation relations, Comm. Math. Phys., 308 (2011), 115–132. https://doi.org/10.1007/s00220-011-1323-9 doi: 10.1007/s00220-011-1323-9
[16]	A. Miyagawa, R. Speicher, A dual and conjugate system for $q$-Gaussians for all $q$, Adv. Math., 413 (2023), 108834. https://doi.org/10.1016/j.aim.2022.108834 doi: 10.1016/j.aim.2022.108834
[17]	M. Kumar, A. Skalski, M. Wasilewski, Full solution of the factoriality question for $q$-Araki-Woods von Neumann algebras via conjugate variables, Comm. Math. Phys., 402 (2023), 157–167. https://doi.org/10.1007/s00220-023-04734-5 doi: 10.1007/s00220-023-04734-5
[18]	M. Bożejko, H. Yoshida, Generalized $q$-deformed Gaussian random variables, Banach Center Publ., 73 (2006), 127–140.
[19]	N. Blitvić, The $(q, t)$-Gaussian process, J. Funct. Anal., 263 (2012), 3270–3305. https://doi.org/10.1016/j.jfa.2012.08.006 doi: 10.1016/j.jfa.2012.08.006
[20]	N. Blitvić, Two-parameter non-commutative Central Limit Theorem, Ann. Inst. Henri Poincaré Probab. Stat., 50 (2014), 1456–1473. https://doi.org/10.1214/13-AIHP550 doi: 10.1214/13-AIHP550
[21]	M. Bożejko, W. Ejsmont, T. Hasebe, Fock space associated to Coxeter groups of type B, J. Funct. Anal., 269 (2015), 1769–1795. https://doi.org/10.1016/j.jfa.2015.06.026 doi: 10.1016/j.jfa.2015.06.026
[22]	M. Bożejko, W. Ejsmont, The double Fock space of type B, SIGMA, 19 (2023), 1–22. https://doi.org/10.3842/SIGMA.2023.040 doi: 10.3842/SIGMA.2023.040
[23]	M. Bożejko, A. Krystek, Ł. Wojakowski, Remarks on the $r$ and $\Delta$ convolutions, Math. Z., 253 (2006), 177–196. https://doi.org/10.1007/s00209-005-0898-2 doi: 10.1007/s00209-005-0898-2
[24]	A. Kula, Convolutions related to $q$-deformed commutativity, Banach Center Publ., 89 (2010), 189–200.
[25]	A. Kula, A limit theorem for the $q$-convolution, Banach Center Publ., 96 (2011), 245–255.
[26]	P. Biane, Some properties of crossings and partitions, Discrete Math., 175 (1997), 41–53. https://doi.org/10.1016/S0012-365X(96)00139-2 doi: 10.1016/S0012-365X(96)00139-2
[27]	M. Anshelevich, S. T. Belinschi, M. Bożejko, F. Lehner, Free infinite divisibility for $q$-Gaussians, Math. Res. Lett., 17 (2010), 905–916. https://doi.org/10.4310/MRL.2010.v17.n5.a8 doi: 10.4310/MRL.2010.v17.n5.a8
[28]	T. Liu, B. Ding, A radial basis function neural network approach for solving a diffusion partial differential equation efficiently, Appl. Math. Comput., 509 (2026), 129651. https://doi.org/10.1016/j.amc.2025.129651 doi: 10.1016/j.amc.2025.129651
[29]	T. Liu, R. Xue, M. Barfeie, A unified framework for high-order compact finite differences using infinitely- and piecewise-smooth RBFs with polynomials, Calcolo, 63 (2026), 18. https://doi.org/10.1007/s10092-026-00684-1 doi: 10.1007/s10092-026-00684-1
[30]	Y. Liu, Y. Li, T. Liu, An RBF-FD method for pricing under the Bates model: handling stochastic volatility and jump processes, Eng. Anal. Bound. Elem., 183 (2026), 106622. https://doi.org/10.1016/j.enganabound.2025.106622 doi: 10.1016/j.enganabound.2025.106622
[31]	T. Liu, R. Xue, A convergent multi-step efficient iteration method to solve nonlinear equation systems, J. Appl. Math. Comput., 71 (2025), 2571–2588. https://doi.org/10.1007/s12190-024-02324-9 doi: 10.1007/s12190-024-02324-9
[32]	A. Nica, R. Speicher, Lectures on the combinatorics of free probability, Cambridge: Cambridge University Press, 2006. https://doi.org/10.1017/CBO9780511735127
[33]	A. Alahmade, Algebraic central limit theorems in noncommutative probability, Ph.D. thesis, Ireland, University College Cork, 2022.

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)