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.