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Résumé |
A meandric system of size n is a non-intersecting collection of closed loops in the plane crossing the real line in exactly 2n points (up to continuous deformation). In mathematical physics terms, it can be seen as a loop model on a random lattice. Connected meandric systems are called meanders, and their enumeration is a notorious hard problem in enumerative combinatorics. In this talk, we discuss a different question, raised independently by Goulden--Nica--Puder and Kargin: what is the number of connected components $cc(M_n)$ of a uniform random meandric system of size 2n? We prove that this number grows linear with n, and concentrates around its mean value, in the sense that $cc(M_n)/n$ converges in probability to a constant. Our main tool is the definition of a notion of local convergence for meandric systems, and the identification of the “quenched Benjamini--Schramm†limit of $M_n$. The latter is the so-called infinite noodle, a largely not understood percolation model recently introduced by Curien, Kozma, Sidoravicius and Tournier.
Our main result has also a geometric interpretation, regarding the Hasse diagram $H_n$ of the non-crossing partition lattice $NC(n)$: informally, our result implies that, in $H_n$, almost all pairs of vertices are asymptotically at the same distance from each other. We use here a connection between $H_n$ and meandric systems discovered by Goulden, Nica and Puder.
Based on joint work with Paul Thevenin (University of Vienna).
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