Permutons, meanders, and SLE-decorated Liouville quantum gravity

In 1912 Henri Poincaré asked the following question: “In how many different ways a simple loop in the plane, called meander, can cross a line a specified number of times?”

Despite many efforts, this question remains open after more than a century. In this talk we construct and study the conjectural scaling limit of uniform meanders. More precisely, we present a natural procedure to construct a continuous permutation (i.e. a permuton) from a pair of space-filling Schramm-Loewner evolution (SLE) curves on a Liouville quantum gravity (LQG) surface. Using this procedure, we then explain how one can construct the meandric permuton, which we conjecture (building on some physics works) to describe the scaling limit of uniform meandric permutations, i.e., the permutations induced by meanders. Using the same techniques, one can also recover the skew Brownian permutons, which describe the scaling limit of various types of random pattern-avoiding permutations already studied in the literature.

We prove several results on these permutons. For instance, (1) we show that for any sequence of random permutations which converges to one of the above random permutons, the length of the longest increasing subsequence is sublinear; (2) we prove that the closed support of each of the random permutons in our class has Hausdorff dimension 1.

Based on this joint work with Ewain Gwynne and Xin Sun.

Slides (PDF)