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We study a class of random permutons which can be constructed from a pair of
space-filling Schramm-Loewner evolution (SLE) curves on a Liouville quantum
gravity (LQG) surface. This class includes the skew Brownian permutons
introduced by Borga (2021), which describe the scaling limit of various types
of random pattern-avoiding permutations. Another interesting permuton in our
class is the meandric permuton, which corresponds to two independent SLE$_8$
curves on a $\gamma$-LQG surface with $\gamma = \sqrt{\frac13 \left( 17 – \sqrt{145} \right)}$. Building on work by Di Francesco, Golinelli, and Guitter
(2000), we conjecture that the meandric permuton describes the scaling limit of
uniform meandric permutations, i.e., the permutations induced by a simple loop
in the plane which crosses a line a specified number of times.

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. This proves that the length of the longest increasing subsequence
is sublinear for Baxter, strong-Baxter, and semi-Baxter permutations and leads
to the conjecture that the same is true for meandric permutations. We also
prove that the closed support of each of the random permutons in our class has
Hausdorff dimension one. Finally, we prove a re-rooting invariance property for
the meandric permuton and write down a formula for its expected pattern
densities in terms of LQG correlation functions (which are known explicitly)
and the probability that an SLE$_8$ hits a given set of points in numerical
order (which is not known explicitly). We conclude with a list of open
problems.

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