Probabilistic operator Algebra Seminar: Asymptotics for a class of meandric systems, via the Hasse diagram of $NC(n)$
Seminar | February 12 | 2-4 p.m. | 736 Evans Hall
Alexandru Nica, University of Waterloo
I will present a joint paper with Ian Goulden and Doron Puder (arXiv:1708.05188), concerning a family of diagrammatic objects called meandric systems. These objects have received a substantial amount of interest from mathematical physicists and from combinatorialists, and the study of the number of components of a random meandric system offers some very appealing, yet difficult problems. In particular, we are not aware of any precise results concerning the asymptotic behaviour for $n \rightarrow \infty$ of the expected number of components of a random meandric system of order $n$. It is known that an equivalent description of the framework of meandric systems can be obtained by looking at the Hasse diagrams (i.e. graphs of covers) of the lattices of non-crossing partitions $NC(n)$, and by studying the distance between two random partitions in $NC(n)$. By using this point of view we identify a non-trivial class of meandric systems “with shallow top”, for which we can determine the precise asymptotic for expected number of components. The calculations leading to this result are inspired by free probability methods, specifically by the idea of taking derivative at time $t = 1$ in a semigroup for free additive convolution. Let $c_n$ denote the expected number of components of a general meandric system of order $n$. By using a variation of the methods from the shallow-top case, we prove that $c_n$ is in the interval $[0.17n,0.51n]$ for n large enough. These bounds support the conjecture that $c_n$ follows a regime of “constant times $n$”, where numerical experiments suggest that the constant is around 0.23. If time permits, I will also present another result of the same paper, giving a precise formula for the number of connected meandric systems (a.k.a. “meanders”) of order n which have shallow top. The formula is derived as a consequence of a bijection between meanders with shallow top and a class of trees. The average growth rate for meanders with shallow top comes out smaller than the estimates known for the growth rate for general meanders, confirming the heuristics that a meandric system with shallow top is likely to have more components (and is less likely to be connected) than a general one.