Generating series and matrix models for meandric systems with one shallow side

Abstract

In this article, we investigate meandric systems having one shallow side: the arch configuration on that side has depth at most two. This class of meandric systems was introduced and extensively examined by I. P. Goulden, A. Nica, and D. Puder in 2020. Shallow arch configurations are in bijection with the set of interval partitions. We study meandric systems by using moment-cumulant transforms for non-crossing and interval partitions, corresponding to the notions of free and boolean independence, respectively, in non-commutative probability. We obtain formulas for the generating series of different classes of meandric systems with one shallow side, by explicitly enumerating the simpler, irreducible objects. In addition, we propose random matrix models for the corresponding meandric polynomials, which can be described in the language of quantum information theory, in particular that of quantum channels.

arXiv comment22 pages

References in corpus (9)

• Random quantum channels I: graphical calculus and the Bell state phenomenon
• Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems
• Meander, Folding and Arch Statistics
• Relations between cumulants in noncommutative probability
• Cumulants in noncommutative probability II. Generalized Gaussian random variables
• Gaussianization and eigenvalue statistics for random quantum channels (III)
• Additivity rates and PPT property for random quantum channels
• Enumerating meandric systems with large number of loops
• An operator that relates to semi-meander polynomials via a two-sided q-Wick formula