Baxter permuton and Liouville quantum gravity

Jacopo Borga, Nina Holden, Xin Sun, Pu Yu

Research output: Contribution to journalArticlepeer-review


The Baxter permuton is a random probability measure on the unit square which describes the scaling limit of uniform Baxter permutations. We determine an explicit formula for the density of the expectation of the Baxter permuton. This answers a question of Dokos and Pak (Online J Anal Comb 9:12, 2014). We also prove that all pattern densities of the Baxter permuton are strictly positive, distinguishing it from other permutons arising as scaling limits of pattern-avoiding permutations. Our proofs rely on a recent connection between the Baxter permuton and Liouville quantum gravity (LQG) coupled with the Schramm-Loewner evolution (SLE). The method works equally well for a two-parameter generalization of the Baxter permuton recently introduced by the first author, except that the density is not as explicit. This new family of permutons, called skew Brownian permuton, describes the scaling limit of a number of random constrained permutations. We finally observe that in the LQG/SLE framework, the expected proportion of inversions in a skew Brownian permuton equals π-2θ2π where θ is the so-called imaginary geometry angle between a certain pair of SLE curves.

Original languageEnglish (US)
Pages (from-to)1225-1273
Number of pages49
JournalProbability Theory and Related Fields
Issue number3-4
StatePublished - Aug 2023


  • Baxter permutations
  • Liouville quantum gravity
  • Permutons
  • Schramm-Loewner evolutions

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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