Traffic probability for rectangular random matrices (Q6592974)

This article discusses the generalization of Male's traffic distributions (see [\textit{C. Male}, Traffic distributions and independence: permutation invariant random matrices and the three notions of independence. Providence, RI: American Mathematical Society (AMS) (2020; Zbl 1487.60002)] for an introduction) to the case of rectangular matrix ensembles that are bi-invariant under permutations. The problem is restated using an embedding in square block matrices. Traffic spaces are generalized using the notion of rectangular probabability space introduced in the work of Benaych-Georges (see in particular [\textit{F. Benaych-Georges}, Probab. Theory Relat. Fields 144, No. 3--4, 471--515 (2009; Zbl 1171.15022)]), and the addition of a grading structure to the graphs considered in usual traffic spaces.\N\NConsider \(l\) independent families of block permutation-invariant rectangular matrices \(\mathbf{B_{1}}, \ldots, \mathbf{B_{l}}\), each admitting a limiting infinitesimal traffic distribution in the large dimension limit. The main result of this article allows to deduce the existence of a limiting infinitesimal traffic distribution for the joint family \((\mathbf{B_{1}}, \ldots, \mathbf{B_{l}})\). This is the rectangular analogue of a result of Male for square matrices.\N\NRectangular matrices with structured entries are studied using the notion of clickable cumulants introduced by \textit{Ø. Ryan} [Commun. Math. Phys. 193, No. 3, 595--626 (1998; Zbl 0912.60013)]. Key to the analysis of he distribution of rectangular matrices with structured entries is the new notion of clickable traffic. Finally, the relation to other types of operator independence is discussed.