The skew Brownian permuton: a new universality class for random constrained permutations
Abstract
We construct a new family of random permutons, called skew Brownian permuton, which describes the limits of several models of random constrained permutations. This family is parametrized by two real parameters. For a specific choice of the parameters, the skew Brownian permuton coincides with the Baxter permuton, i.e., the permuton limit of Baxter permutations. We prove that for another specific choice of the parameters, the skew Brownian permuton coincides with the biased Brownian separable permuton, a oneparameter family of permutons previously studied in the literature as the limit of uniform permutations in substitutionclosed classes. This brings two different limiting objects under the same roof, identifying a new larger universality class. The skew Brownian permuton is constructed in terms of flows of solutions of certain stochastic differential equations (SDEs) driven by twodimensional correlated Brownian excursions in the nonnegative quadrant. We call these SDEs skew perturbed Tanaka equations because they are a mixture of the perturbed Tanaka equations and the equations encoding skew Brownian motions. We prove existence and uniqueness of (strong) solutions for these new SDEs. In addition, we show that some natural permutons arising from Liouville quantum gravity spheres decorated with two SchrammLoewner evolution curves are skew Brownian permutons and such permutons cover almost the whole range of possible parameters. Some connections between constrained permutations and decorated planar maps have been investigated in the literature at the discrete level; this paper establishes this connection directly at the continuum level. Proving the latter result, we also give an SDE interpretation of some quantities related to SLEdecorated Liouville quantum gravity spheres.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 DOI:
 10.48550/arXiv.2112.00156
 arXiv:
 arXiv:2112.00156
 Bibcode:
 2021arXiv211200156B
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Combinatorics
 EPrint:
 New version including referee's corrections, accepted for publication in Proceedings of the London Mathematical Society