Meanders, meandric permutations, and meandric systems

The images on this page are taken from this work on meanders and this work on meandric systems.

Meanders and meandric permutations

Left: Two large uniform meanders of size 256 and 2048. Right: The plots of the two corresponding meandric permutations. These simulations are obtained using this paper‘s Markov chain Monte Carlo algorithm.

Meandric systems

Simulation of a uniform meandric system with boundary of size 1000000 (see Section 7.4 of the papervfor a precise definition and for the details of simulations). The left picture shows the corresponding arc diagrams. The right picture shows the associated planar map, embedded in the disk via the Tutte embedding, together with some of the loops of the meandric system. The largest 300 loops (in terms of number of vertices) are each shown in color, as indicated by the color bar. Smaller loops and edges between consecutive vertices of the real line are shown in gray. Note that the distribution of colors in the arc diagram picture is rather chaotic – this is consistent with the fact that the meandric system loops are a complicated functional of the arc diagrams. According to our conjectures, the embedded planar map together with the path induced by the real line, and the loops, should converge to √ 2-LQG decorated by SLE8 and CLE6.