Alexandru Nica, Department of Pure Mathematics, University of Waterloo
"Free probabilistic aspects of meandric systems"
I will consider a family of diagrammatic objects (well-known to mathematical physicists and to combinatorialists) which go under the name of "meandric systems". These objects offer some very appealing, yet difficult problems -- in particular, denoting by $E_n$ the expected number of components of a random meandric system of order $n$, there are no precise results concerning the asymptotic behaviour of $E_n$ for large values of $n$. Numerical experiments suggest the conjecture that the limit of $E_n / n$ should exist, with a value around 0.23. In this talk I will present a result obtained in joint work with Ian Goulden and Doron Puder, giving some evidence in favour of the above conjecture. Quite interestingly, our result is intimately related to the combinatorial side of free probability, in particular to a basic notion of "free additive convolution" (for probability measures on the real line) which is used in free probability and will be reviewed as part of the talk.
MC 5417