# Permutons, meanders, and SLE-decorated Liouville quantum gravity

Jacopo Borga Stanford University Ewain Gwynne University of Chicago Xin Sun
University of Pennsylvania

###### Abstract

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 SLE8
curves on a $\gamma$-LQG surface with
$\gamma=\sqrt{\frac{1}{3}\mathopen{}\mathclose{{}\left(17-\sqrt{145}}\right)}$.
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 SLE8 hits a given set
of points in numerical order (which is not known explicitly). We conclude with
a list of open problems.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Permutons constructed from SLEs and LQG
    3. 1.3 Meanders, meandric permutations, and their conjectured connection to SLEs and LQG
    4. 1.4 Skew Brownian permutons
    5. 1.5 Main results
  2. 2 Preliminaries
    1. 2.1 Gaussian free field
    2. 2.2 Liouville quantum gravity
    3. 2.3 The quantum sphere
    4. 2.4 Space-filling SLE
    5. 2.5 Basic properties of permutons constructed from SLEs and LQG
  3. 3 Longest increasing subsequence is sublinear
    1. 3.1 Longest increasing subsequences and monotone sets for permutons
    2. 3.2 Monotone sets for permutons constructed from SLEs and LQG
  4. 4 Dimension of the support is one
    1. 4.1 Core argument
    2. 4.2 Proofs of LQG metric estimates
  5. 5 Re-rooting invariance
    1. 5.1 Re-rooting invariance for SLE8
    2. 5.2 Re-rooting invariance for SLE8-decorated $\gamma$-LQG
  6. 6 Physics background for the meander conjecture
    1. 6.1 Meanders, Hamiltonian cycles, and spanning trees
    2. 6.2 Positive temperature O($n$) loop model and its scaling limit
    3. 6.3 Fully packed O($n$) loop model and its scaling limit
    4. 6.4 Fully packed O($n$) loop model on random planar maps
    5. 6.5 Meanders and the fully packed O$(n\times m)$ loop model
    6. 6.6 Convergence of the cyclic meandric permutation to the identity
  7. 7 Open problems

##  1 Introduction

###  1.1 Overview

A permuton is a probability measure ${\boldsymbol{\pi}}$ on $[0,1]^{2}$ whose
one-dimensional marginals are each equal to Lebesgue measure on $[0,1]$.
Permutons are of interest because they describe the scaling limits of various
types of random permutations (see [Bor21c, Section 1.1] and the references
therein) and they identify a natural space where one can study large
deviations principles for permutations (as done in [Muk16, KKRW20, BDMW22]).
To be more precise, each permutation is associated to a permuton as follows.

######  Definition 1.1.

For a permutation $\sigma$ on $[1,n]\cap\mathbbm{Z}$, we define the size of
$\sigma$ by $|\sigma|=n$. We use the one-line notation to write permutations,
that is, if $\sigma$ is a permutation of size $n$ then we write
$\sigma=\sigma(1),\dots,\sigma(n)$. We define the permuton associated with
$\sigma$ to be the measure ${\boldsymbol{\pi}}_{\sigma}$ on $[0,1]^{2}$ which
is equal to $|\sigma|$ times Lebesgue measure on

$\bigcup_{j=1}^{|\sigma|}\mathopen{}\mathclose{{}\left[\frac{j-1}{|\sigma|},\frac{j}{|\sigma|}}\right]\times\mathopen{}\mathclose{{}\left[\frac{\sigma(j)-1}{|\sigma|},\frac{\sigma(j)}{|\sigma|}}\right].$

Figure 1:  The permuton ${\boldsymbol{\pi}}_{\sigma}$ associated with the
permutation $\sigma=1,8,3,2,4,6,7,5$ is equal to 8 times the Lebesgue measure
on the gray subset of $[0,1]^{2}$.

See Figure 1 for an illustration. We equip the space of permutons with the
usual weak topology on measures, and we say that a sequence of permutations
converges to a permuton ${\boldsymbol{\pi}}$ if their associated permutons
converge weakly. We refer to [Bor21a, Remark 2.1.2] for a brief history of the
theory of permutons.

In this paper, we will study a certain class of random permutons which we
define precisely in Section 1.2. We will be especially interested in two
special cases.

  * •

The meandric permuton is a random permuton which we define in Definition 1.6
and which we conjecture describes the scaling limit of uniform meandric
permutations, i.e., the permutations induced by a pair of intersecting Jordan
curves (see Section 1.3 for more details).

  * •

The skew Brownian permutons are a two-parameter family of random permutons
introduced by Borga in [Bor21c] which describe the scaling limits of various
types of random pattern-avoiding permutations (see Section 1.4 for more
details).

For various subsets of our class of permutons, including the two special cases
listed above, we will show that the length of the longest increasing
subsequence for any sequence of permutations converging to the permuton is
sublinear (Theorem 1.12); and that the dimension of the closed support of the
permuton is one (Theorem 1.19). We will also establish a re-rooting property
for the meandric permuton (Theorem 1.21) and a formula for its pattern
densities in terms of hitting probabilities for SLE8 and the correlation
functions of the LQG area measure (Theorem 1.22).

The common feature of all the permutons we consider is that they can be
represented in terms of a Liouville quantum gravity (LQG) surface decorated by
a pair of coupled Schramm-Loewner evolution (SLE) curves. There is a vast
literature concerning SLE and LQG, but not much prior knowledge of this
literature is needed to understand this paper. All of the necessary background
knowledge about SLE and LQG is explained in Section 2. Our proofs will only
use relatively simple statements about these objects which we re-state as
necessary and which can be taken as black boxes.

Acknowledgments. We thank Rick Kenyon for helpful discussions on the uniform
spanning tree. We thank Tony Guttmann and Iwan Jensen for helpful discussions
on numerical simulations. We thank Zhihan Li for coding the Markov chain Monte
Carlo algorithm from [HT11] which was used to obtain the pictures in Figure 4
and the numerical simulations for Conjecture 1.17. Part of the work on this
paper was carried out during visits by J.B. to Chicago and Penn, by E.G. to
Stanford, and by X.S. to Stanford. We thank these institutions for their
hospitality. E.G. was partially supported by a Clay research fellowship. X.S.
was supported by the NSF grant DMS-2027986 and the NSF Career Award 2046514.

###  1.2 Permutons constructed from SLEs and LQG

Let us now define the family of permutons which we will consider. Fix
parameters $\gamma\in(0,2)$ and $\kappa_{1},\kappa_{2}>4$. Let $h$ be a random
generalized function on $\mathbbm{C}$ corresponding to a singly marked unit
area $\gamma$-Liouville quantum sphere, and let $\mu_{h}$ be its associated
$\gamma$-LQG area measure. We will review the definitions of these objects in
Sections 2.2 and 2.3, but for the time being the reader can just think of
$\mu_{h}$ as a random, non-atomic, Borel probability measure on $\mathbbm{C}$
which assigns positive mass to every open subset of $\mathbbm{C}$.

Independently from $h$, let $(\eta_{1},\eta_{2})$ be a random pair consisting
of a whole-plane space-filling SLE${}_{\kappa_{1}}$ curve and a whole-plane
space-filling SLE${}_{\kappa_{2}}$ curve, each going from $\infty$ to
$\infty$. We will review the definition of these random curves in Section 2.4,
but for the time being the reader can just think of $(\eta_{1},\eta_{2})$ as a
pair of random non-self-crossing space-filling curves in $\mathbbm{C}$ which
each visit almost every point of $\mathbbm{C}$ exactly once. We parametrize
each of $\eta_{1}$ and $\eta_{2}$ by $\mu_{h}$-mass, i.e., in such a way that
$\mu_{h}(\eta_{1}([0,t]))=\mu_{h}(\eta_{2}([0,t]))=t$ for each $t\in[0,1]$.

We emphasize that the coupling of our two SLE curves, viewed modulo time
parametrization, is arbitrary. We will be interested in cases where the two
curves determine each other as well as the case where the two curves are
independent.

Let $\psi:[0,1]\rightarrow[0,1]$ be a Lebesgue measurable function such that

$\eta_{1}(t)=\eta_{2}(\psi(t)),\quad\forall t\in[0,1].$ (1.1)

The permuton associated with $(h,\eta_{1},\eta_{2})$ is the random probability
measure ${\boldsymbol{\pi}}$ on $[0,1]^{2}$ defined by

${\boldsymbol{\pi}}(A)={\boldsymbol{\pi}}_{\eta_{1},\eta_{2}}(A)=\operatorname{Leb}\mathopen{}\mathclose{{}\left\\{t\in[0,1]:(t,\psi(t))\in
A}\right\\},\quad\text{$\forall$ Borel sets $A\subset[0,1]^{2}$},$ (1.2)

where $\operatorname{Leb}$ denotes Lebesgue measure. Using the fact that
$\mu_{h}$-a.e. point in $\mathbbm{C}$ is hit only once by each of $\eta_{1}$
and $\eta_{2}$, is easy to see that the definition of ${\boldsymbol{\pi}}$
does not depend on the choice of $\psi$ (see Lemma 2.8).

Before we state our main results, we discuss the two special cases of the
construction in this subsection which are particularly interesting. See Figure
2 for a Venn diagram of the different permutons we will consider in this
paper.

Figure 2:  Venn diagram of the different classes of random permutons
considered in this paper. The outer grey oval represents all random permutons
of the type constructed in (1.2). The meandric permuton and the skew Brownian
permutons are special cases of this construction. In particular, the meandric
permuton falls in the subclass of permutons constructed from independent SLEs.

###  1.3 Meanders, meandric permutations, and their conjectured connection to
SLEs and LQG

######  Definition 1.2.

A meander of size $n\in\mathbbm{N}$ is a simple loop $\ell$ (i.e., a
homeomorphic image of the circle) in $\mathbbm{R}^{2}$ which crosses the real
line $\mathbbm{R}$ exactly $2n$ times and does not touch the line without
crossing it, viewed modulo orientation-preserving homeomorphisms from
$\mathbbm{R}^{2}$ to $\mathbbm{R}^{2}$ which take $\mathbbm{R}$ to
$\mathbbm{R}$.

See the left-hand sides of Figures 3 and 4 for some illustrations of meanders.
A meander of size $n$ can equivalently be thought of as a pair of Jordan
curves in the sphere which intersect exactly $2n$ times and which do not
intersect without crossing, viewed modulo orientation-preserving
homeomorphisms from the sphere to itself. The correspondence with Definition
1.2 is obtained by viewing $\mathbbm{R}$ as a curve in the Riemann sphere. See
the right-hand side of Figure 3.

Figure 3: Left: A meander of size $6$, with the intersection points of the
loop and the line labeled by the order in which they are hit by the loop (in
black) and the line (in red) starting at the blue vertex. The associated
meandric permutation is $\sigma_{\ell}=1,4,3,2,5,12,7,8,9,10,11,6$ (in one-
line notation); see Definition 1.5 below. Right: The same meander of size $6$
seen as a pair of Jordan curves in the sphere.

The study of meanders dates back to at least the work of Poincaré [Poi12],
although the term “meander” was first coined by Arnold [Arn88]. Meanders are
of interest in physics and computational biology as models of polymer folding
[DFGG97]. There is a vast mathematical literature devoted to the enumeration
of various types of meanders, which has connections to many different
subjects, from combinatorics to theoretical physics and more recently to the
geometry of moduli spaces [DGZZ20]. See [Zvo21] for a brief recent survey of
this literature and [La 03] for a more detailed account. Additionally,
meanders are the connected components of _meandric systems_ , recently studied
in [CKST19, Kar20, GNP20, FT22, FN19].

The problem of enumerating _all_ meanders of size $n$ is open and seemingly
quite difficult. But, it was conjectured by Di Francesco, Golinelli, and
Guitter [DFGG00] that the number of meanders of size $n$ behaves
asymptotically like $C\cdot A^{n}n^{-\alpha}$, where
$\alpha=(29+\sqrt{145})/12$. This conjecture was later numerically tested by
Jensen and Gutmann [JG00], where they also proposed the numerical estimate of
$A\approx 12.26$. We will discuss this conjecture in more detail in Section 6;
see also Remark 2.6. The best known rigorous bounds for the constant $A$ are
$11.380\leq A\leq 12.901$, as proved in [AP05].

In this paper, we will be interested in the scaling limit of a uniform sample
$\ell_{n}$ from the set of all meanders of size $n$. One way to formulate such
a scaling limit is as follows. We associate with $\ell_{n}$ the planar map
$M_{n}$ whose vertices are the $2n$ intersection points of the loop $\ell_{n}$
with the real line, whose edges are the segments of the loop or the line
$\mathbbm{R}$ between these intersection points (where we consider the two
infinite rays of the line $\mathbbm{R}$ as being an edge from the leftmost to
the rightmost intersection point), and whose faces are the connected
components of $\mathbbm{R}^{2}\setminus(\ell_{n}\cup\mathbbm{R})$. The planar
map $M_{n}$ is equipped with two Hamiltonian paths
$P_{n}^{1},P_{n}^{2}:[0,2n]\cap\mathbbm{Z}\rightarrow\\{\text{vertices of
$M_{n}$}\\}$, one of which is induced by the left-right ordering of
$\mathbbm{R}$ and one of which is induced by $\ell_{n}$ (these are the black
and red paths in Figure 3). We view each of these paths as starting and ending
at the leftmost intersection point of $\ell_{n}$ with $\mathbbm{R}$.

To talk about convergence, we can, e.g., view $(M_{n},P_{n}^{1},P_{n}^{2})$ as
a curve-decorated metric measure space (equipped with the graph metric and the
counting measure on vertices) and ask whether it has a scaling limit with
respect to the generalization of the Gromov-Hausdorff topology for curve-
decorated metric measure spaces [GM17]. As we will explain in Section 6, the
arguments of [DFGG00] lead to the following conjecture.

######  Conjecture 1.3.

Let $(M_{n},P_{n}^{1},P_{n}^{2})$ be the random planar map decorated by two
Hamiltonian paths associated with the uniform meander $\ell_{n}$ as above.
Then $(M_{n},P_{n}^{1},P_{n}^{2})$ converges under an appropriate scaling
limit to a Liouville quantum gravity (LQG) sphere with matter central charge
${\mathbf{c}_{\mathrm{M}}}=-4$ (equivalently, from (2.12), with coupling
constant
$\gamma=\sqrt{\frac{1}{3}\mathopen{}\mathclose{{}\left(17-\sqrt{145}}\right)}$),
decorated by a pair of independent whole-plane SLE8 curves from $\infty$ to
$\infty$, each parametrized by LQG mass with respect to the quantum sphere.

An equivalent formulation of the conjecture is that
$(M_{n},P_{n}^{1},P_{n}^{2})$ should be in the same universality class as a
uniform 3-tuple consisting of a planar map decorated by two spanning trees
(and their associated discrete Peano curves). See Section 6 for more details
about the reasons behind the conjecture. We will not make any direct progress
toward a proof of Conjecture 1.3 in this paper. Rather, we will study the
conjectural limiting object directly.

######  Remark 1.4.

Most results concerning SLE-decorated Liouville quantum gravity are specific
to the case when $\kappa\in\\{\gamma^{2},16/\gamma^{2}\\}$ [She16a, DMS21]
(see [GHS19] for a survey of such results). In the setting of Conjecture 1.3,
the values of $\gamma$ and $\kappa$ are _mismatched_ in the sense that
$\kappa\notin\\{\gamma^{2},16/\gamma^{2}\\}$. This case is much less well
understood than the matched case.

Another possible perspective on the scaling limit of meanders is to look at
the permutation associated with a meander $\ell$.

######  Definition 1.5.

For a meander $\ell$ of size $n$, the corresponding meandric permutation
$\sigma_{\ell}$ is the permutation of $[1,2n]\cap\mathbbm{Z}$ obtained as
follows. Order the $2n$ intersection points of $\ell$ with $\mathbbm{R}$ from
left to right. Then, consider the second order in which these intersection
points are hit by $\ell$ when we start $\ell$ from the leftmost intersection
point and traverse it counterclockwise. Then $\sigma_{\ell}(i)=j$ if some
intersection point is the $i$-th visited point by the first order and the
$j$-th visited point by the second one.

Figure 4: Left: Two large uniform meanders $\ell$ of size 256 and 2048. Right:
The plots of the two corresponding meandric permutations $\sigma_{\ell}$; the
blue dots correspond to the values
$(i,\sigma_{\ell}(i))_{i=1,\dots,|\sigma_{\ell}|}$. These simulations are
obtained using the Markov chain Monte Carlo algorithm from [HT11].

See Figure 3 for an example and the right-hand side of Figure 4 for the plots
of two large uniform meandric permutations. Note that any meandric permutation
fixes 1. It is easily verified that the mapping $\ell\mapsto\sigma_{\ell}$ is
a bijection from the set of meanders of size $n$ to the set of meandric
permutations of size $2n$. The study of meandric permutations dates back to at
least the paper [Ros84], where they are called “planar permutations”. We note
that meandric permutations are defined differently in various other
literature; see Section 1.3.1 just below for further explanations.

Conjecture 1.3 leads naturally to the following definition and conjecture.

######  Definition 1.6.

The meandric permuton is the permuton ${\boldsymbol{\pi}}$ defined as in (1.2)
in the case when
$\gamma=\sqrt{\frac{1}{3}\mathopen{}\mathclose{{}\left(17-\sqrt{145}}\right)}$,
$\kappa_{1}=\kappa_{2}=8$, and $\eta_{1}$ and $\eta_{2}$ are independent,
viewed as curves modulo time parametrization.

######  Conjecture 1.7.

For $n\in\mathbbm{N}$, let $\sigma_{n}$ be a uniform sample from the set of
meandric permutations of size $2n$ (i.e., the permutation associated with a
uniform meander of size $n$). Then the associated permuton
${\boldsymbol{\pi}}_{\sigma_{n}}$ (Definition 1.1) converges in law to the
meandric permuton.

It is possible that Conjecture 1.7 is easier to prove than Conjecture 1.3.
This is because permutons are in some sense simpler objects than SLE-decorated
LQG and because the set of permutons is compact with respect to the weak
topology (so tightness is automatic). Nevertheless, we do not know of a
strategy for proving either Conjecture 1.3 or 1.7 (see, however, Problem 7.1).

We plan to prove in future work that the field $h$ and the SLE8 curves
$\eta_{1}$ and $\eta_{2}$, viewed modulo conformal and anticonformal
automorphisms of $\mathbbm{C}$, are a.s. given by (somewhat explicit)
measurable functions of the meandric permuton ${\boldsymbol{\pi}}$.
Consequently, Conjecture 1.7 can still be thought of as a form of convergence
of meanders toward SLE-decorated LQG.

####  1.3.1 Cyclic meandric permutations

In various other literature, e.g., [La 03], the meandric permutation is
instead defined as follows. Label the intersection points of the meander
$\ell$ from left to right as $1,\dots,2n$. For $j=1,\dots,2n$, let
$\tau_{\ell}(j)$ be the index of the intersection point hit by the loop
(traversed counterclockwise) immediately after it hits $j$. Then $\tau_{\ell}$
is a cycle of length $2n$. To distinguish $\tau_{\ell}$ from the meandric
permutation $\sigma_{\ell}$ of Definition 1.5, we call $\tau_{\ell}$ the
cyclic meandric permutation.

If we express our meandric permutation $\sigma_{\ell}$ in one-line notation
(as in Figure 3), then the same expression, interpreted as a permutation
written in cyclic notation, is equal to $\tau_{\ell}$. The permutations
$\tau_{\ell}$ and $\sigma_{\ell}$ are related by

$\sigma_{\ell}(j)=\tau_{\ell}^{j-1}(1)\quad\text{and}\quad\tau_{\ell}(j)=\sigma_{\ell}(\sigma_{\ell}^{-1}(j)+1),\quad\forall
j=1,\dots,2n.$ (1.3)

One reason why $\sigma_{\ell}$ is more natural than $\tau_{\ell}$ from our
perspective is the following conjecture.

######  Conjecture 1.8.

Let $\ell_{n}$ be a uniform meander of size $n$ and let $\tau_{\ell_{n}}$ be
its associated cyclic meandric permutation. The permuton associated with
$\tau_{\ell_{n}}$ converges in law to the identity permuton (i.e., the uniform
measure on the diagonal in $[0,1]^{2}$) as $n\rightarrow\infty$.

See Section 6.6 for a heuristic justification of Conjecture 1.8 using
Conjecture 1.3.

###  1.4 Skew Brownian permutons

The skew Brownian permutons are a two-parameter family of random permutons,
indexed by $(\rho,q)\in(-1,1)\times(0,1)$, which were first introduced in
[Bor21c]. The original construction of the skew Brownian permuton involves a
pair of correlated Brownian excursions with correlation $\rho$. The parameter
$q$ controls, roughly speaking, the tendency of the permuton to be increasing.
This can be made precise by looking at the probability that a pair of points
$(X_{1},Y_{1}),(X_{2},Y_{2})$ sampled from the permuton satisfy $X_{1}<X_{2}$
and $Y_{1}>Y_{2}$ [BHSY22, Proposition 1.14]. The case $q=1/2$ is symmetric,
in the sense that this probability is $1/2$. We will not need the Brownian
motion construction of the skew Brownian permuton in this paper.

It is shown in [Bor21c, Theorem 1.17] that for each choice of parameters
$(\rho,q)\in(-1,1)\times(0,1)$ for the skew Brownian permuton, there exists
$\gamma\in(0,2)$ and a coupling of two whole-plane space-filling
SLE${}_{\kappa=16/\gamma^{2}}$ curves from $\infty$ to $\infty$ such that the
permuton (1.2) coincides with the skew Brownian permuton. See Theorem 2.7
below for a precise statement. This representation has previously been used to
prove various properties of the skew Brownian permuton in [BHSY22].

One of the main reasons to study skew Brownian permutons is that they describe
the scaling limits of various types of pattern-avoiding random permutations,
for example the following.

######  Definition 1.9.

Let $\sigma$ be a permutation of size $n$.

  * •

$\sigma$ is semi-Baxter if there does not exist $1\leq i<j<k\leq n$ such that
$\sigma(j+1)<\sigma(i)<\sigma(k)<\sigma(j)$.

  * •

$\sigma$ is Baxter if $\sigma$ is semi-Baxter and additionally there does not
exist $1\leq i<j<k\leq n$ such that
$\sigma(j)<\sigma(k)<\sigma(i)<\sigma(j+1)$.

  * •

$\sigma$ is strong-Baxter if $\sigma$ is Baxter and additionally there does
not exist $1\leq i<j<k\leq n$ such that
$\sigma(j+1)<\sigma(k)<\sigma(i)<\sigma(j)$.

We have the following convergence results toward skew Brownian permutons.

  * •

The permutons associated with uniform semi-Baxter permutations converge in law
to the skew Brownian permuton with $\rho=-(1+\sqrt{5})/4$ and $q=1/2$
[Bor21b].

  * •

The permutons associated with uniform Baxter permutations converge in law to
the skew Brownian permuton with $\rho=-1/2$ and $q=1/2$ [BM22] (this special
case of the skew Brownian permuton is sometimes called the Baxter permuton).

  * •

The permutons associated with uniform strong-Baxter permutations converge in
law to the skew Brownian permuton with parameters $\rho\approx-0.2151$ and
$q\approx 0.3008$ [Bor21b].

######  Remark 1.10.

The boundary case $\rho=1$, $q\in(0,1)$ for the skew Brownian permuton
corresponds to the Brownian separable permuton [BBF+18] and biased versions
thereof [BBF+20, BBFS20], as proved in [Bor21c, Theorem 1.12]. We will not
consider the case $\rho=1$ in the present paper since the representation in
terms of SLE and LQG from [Bor21c, Theorem 1.17] does not work the same way in
this case. However, some of the results in this paper have already been proven
for the Brownian separable permuton by other methods. It is conjectured in
[Bor21c, Section 1.6, Item 7] that the Brownian separable permuton can be
described in terms of SLE and LQG using the critical mating of trees results
from [AHPS21].

###  1.5 Main results

The main results of this paper and their proofs (given in Sections 3 through
5) can be read independently from each other.

####  1.5.1 Longest increasing subsequence is sublinear

Assume that we are in the setting of Section 1.2. We allow $\gamma\in(0,2)$
and $\kappa_{1},\kappa_{2}>4$ to be arbitrary, but we require that the joint
distribution of the SLE curves $(\eta_{1},\eta_{2})$, viewed as curves modulo
time parametrization, satisfies one of the following two conditions.

  * •

$\eta_{1}$ and $\eta_{2}$ are independent viewed modulo time parametrization
(as is the case in the definition of the meandric permuton); or

  * •

$\kappa_{1}=\kappa_{2}$ and $(\eta_{1},\eta_{2})$ are coupled together as in
the SLE / LQG description of the skew Brownian permuton111We highlight that we
do not require that $\kappa_{i}=16/\gamma^{2}$ but only that
$\kappa_{1}=\kappa_{2}$. for some choice of $(\rho,q)\in(-1,1)\times(0,1)$
(Theorem 2.7).

Let ${\boldsymbol{\pi}}$ be the associated permuton as in (1.2). Note that
${\boldsymbol{\pi}}$ can be either the meandric permuton or the skew Brownian
permuton. In this subsection we will consider the longest increasing
subsequence for random permutations which converge to ${\boldsymbol{\pi}}$.

######  Definition 1.11.

For a permutation $\sigma$, we define the length of the longest increasing
(resp. decreasing) subsequence $\operatorname{LIS}(\sigma)$ (resp.
$\operatorname{LDS}(\sigma)$) to be the maximal cardinality of a set
$L\subset[1,|\sigma|]\cap\mathbbm{Z}$ such that the restriction of $\sigma$ to
$L$ is monotone increasing (resp. decreasing).

There is a huge literature devoted to the asymptotic behavior of
$\operatorname{LIS}(\sigma_{n})$ and $\operatorname{LDS}(\sigma_{n})$ for
various types of large random permutations $\sigma_{n}$. For uniform random
permutations $\sigma_{n}$, the question of investigating
$\operatorname{LIS}(\sigma_{n})$ was raised in the 1960s by Ulam [Ula61]. In
this case, one has $\operatorname{LIS}(\sigma_{n})\sim 2\sqrt{n}$ [Ham72,
VK77, LS77]. The strongest known result is due to Dauvergne and Virág [DV21],
who showed that the scaling limit of the longest increasing subsequence in a
uniform permutation is the _directed geodesic_ of the _directed landscape_.
The study of $\operatorname{LIS}(\sigma_{n})$ is connected with many other
problems in combinatorics and probability theory, such as last passage
percolation and random matrix theory; we refer the reader to the book of Romik
[Rom15] for an overview.

In recent years, many extensions beyond uniform permutations have been
considered:

  * •

when $\sigma_{n}$ is a uniform random pattern-avoiding permutation [DHW03,
MY17, MY20, BBD+21];

  * •

when $\sigma_{n}$ is conjugacy-invariant with few cycles, in particular
_Ewens-distributed_ [SK18];

  * •

when $\sigma_{n}$ follows the _Mallows distribution_ , or is a product of such
random permutations ([Jin19] and references therein).

For the first two items the results are far from being complete, and our next
result is a new contribution towards the understanding of the behavior of
$\operatorname{LIS}(\sigma_{n})$ for random permutations $\sigma_{n}$ which
are not chosen uniformly from all possibilities.

######  Theorem 1.12.

Assume that $(\eta_{1},\eta_{2})$ satisfy the hypotheses at the beginning of
this subsection and let ${\boldsymbol{\pi}}$ be as in (1.2). Let
$\\{\sigma_{n}\\}_{n\in\mathbbm{N}}$ be a sequence of random permutations of
size $|\sigma_{n}|\rightarrow\infty$ whose associated permutons
${\boldsymbol{\pi}}_{\sigma_{n}}$ converge in law to ${\boldsymbol{\pi}}$ with
respect to the weak topology. Then the longest increasing subsequence of
$\sigma_{n}$ is sublinear in $|\sigma_{n}|$, i.e.,
$\operatorname{LIS}(\sigma_{n})/|\sigma_{n}|\rightarrow 0$ in probability. The
same is true for the longest decreasing subsequence.

Theorem 1.12 will be a consequence of a general condition on a permuton which
implies that any permutations converging to it have sublinear longest
increasing subsequences (Proposition 3.2). We will verify this condition in
our setting in Section 3, via a surprisingly simple (3-page) argument using
SLE and LQG.

The analog of Theorem 1.12 for the biased Brownian separable permuton (which
corrresponds to the skew Brownian permuton in the limiting case $\rho=1$) was
proven in [BBD+21, Theorem 1.10], using very different arguments from the ones
in this paper. As an immediate corollary of Theorem 1.12 and the results of
[Bor21c] (see Theorem 2.7), we obtain the following.

######  Corollary 1.13.

Let ${\boldsymbol{\pi}}$ be the skew Brownian permuton with parameters
$\rho\in(-1,1)$ and $q\in(0,1)$. Let $\\{\sigma_{n}\\}_{n\in\mathbbm{N}}$ be a
sequence of random permutations of size $|\sigma_{n}|\rightarrow\infty$ whose
associated permutons ${\boldsymbol{\pi}}_{\sigma_{n}}$ converge in law to
${\boldsymbol{\pi}}$ with respect to the weak topology. Then
$\operatorname{LIS}(\sigma_{n})/|\sigma_{n}|\rightarrow 0$ and
$\operatorname{LDS}(\sigma_{n})/|\sigma_{n}|\rightarrow 0$ in probability.

We now record the implication of Corollary 1.13 for some particular random
permutations which are known to converge to the skew Brownian permuton (see
Section 1.4).

######  Corollary 1.14.

For $n\in\mathbbm{N}$, let $\sigma_{n}$ be a uniform sample from the set of
Baxter permutations of size $n$ (Definition 1.9). Then
$\operatorname{LIS}(\sigma_{n})/n\rightarrow 0$ and
$\operatorname{LDS}(\sigma_{n})/n\rightarrow 0$ in probability. The same is
true for strong-Baxter permutations and semi-Baxter permutations.

Note that not all models of uniform pattern-avoiding permutations have
sublinear longest increasing subsequences; see, e.g., [DHW03, MY17, MY20] for
examples where the longest increasing subsequence is known to be linear.

######  Remark 1.15.

The study of the length of the longest increasing subsequence in a Baxter
permutation $\operatorname{LIS}(\sigma_{n})$ is also motivated by its
connection with the the length of the longest directed path
$\operatorname{LDP}(m_{n})$ from the the source to the sink of a uniform
bipolar orientation $m_{n}$ with $n$ edges. Recall that a bipolar orientation
is a planar map equipped with an acyclic orientation of the edges with exactly
one source (i.e. a vertex with only outgoing edges) and one sink (i.e. a
vertex with only incoming edges), both on the outer face. Uniform bipolar
orientations have been studied, e.g., in [KMSW19, GHS16, BM22].

Building on a bijection between Baxter permutations and bipolar orientations,
introduced by Bonichon, Bousquet-Mélou and Fusy [BBMF11], it is possible to
show that $\operatorname{LIS}(\sigma_{n})$ and $\operatorname{LDP}(m_{n})$
have the same law. Therefore, Corollary 1.14 implies that
$\operatorname{LDP}(m_{n})$ is sublinear.

The study of $\operatorname{LDP}(m_{n})$ can be interpreted as a model of last
passage percolation in random geometry, which, to the best of our knowledge,
has been not investigated previously.

We do not know that uniform meandric permutations converge to the meandric
permuton, but due to Theorem 1.12, the following statement is implied by
Conjecture 1.7.

######  Conjecture 1.16.

Let $\sigma_{n}$ be a uniform meandric permutation of size $2n$. Then
$\operatorname{LIS}(\sigma_{n})/n\rightarrow 0$ and
$\operatorname{LDS}(\sigma_{n})/n\rightarrow 0$ in probability.

It is also of substantial interest to prove non-trivial _lower_ bound for
$\operatorname{LIS}(\sigma_{n})$ and $\operatorname{LDS}(\sigma_{n})$, for
appropriate families of random permutations
$\\{\sigma_{n}\\}_{n\in\mathbbm{N}}$ which converge in law to the permutons
considered in this paper. A classical theorem of Erdös-Szekeres [ES35] shows
that for any permutation $\sigma$, one has
$\operatorname{LIS}(\sigma)\times\operatorname{LDS}(\sigma)\geq|\sigma|$.
Hence, either $\operatorname{LIS}(\sigma)$ or $\operatorname{LDS}(\sigma)$ (or
both) must be at least $\lfloor\sqrt{|\sigma|}\rfloor$.

We expect that for the random permutations considered in Corollary 1.14, both
$\operatorname{LIS}(\sigma_{n})$ and $\operatorname{LDS}(\sigma_{n})$ should
be of order $n^{\alpha}$ for some $\alpha\in(1/2,1)$. By Remark 1.15, similar
statements should also hold for the length of the longest directed path from
the the source to the sink of a uniform bipolar orientation. In the case of
uniform meandric permutations, we have a more precise guess for the growth
rate of $\operatorname{LIS}(\sigma_{n})$ and $\operatorname{LDS}(\sigma_{n})$
coming from numerical simulations.222The numerical simulations used for
Conjecture 1.17 were obtained through the implementation of the Markov chain
Monte Carlo approach from [HT11].

######  Conjecture 1.17.

Let $\sigma_{n}$ be a uniform meandric permutation of size $2n$. Then there
exists $\alpha\approx 0.69$ such that
$\operatorname{LIS}(\sigma_{n})=n^{\alpha+o(1)}$ and
$\operatorname{LDS}(\sigma_{n})=n^{\alpha+o(1)}$ with probability tending to 1
as $n\to\infty$.

We do not have a guess for the precise value of $\alpha$. In the settings of
both Corollary 1.14 and 1.17, we are not currently able to prove any lower
bounds for $\operatorname{LIS}(\sigma_{n})$ and
$\operatorname{LDS}(\sigma_{n})$ which are significantly better than the
trivial bound $n^{1/2}$ which comes from Erdös-Szekeres. See Problems 7.2 and
7.3 for some relevant open problems.

####  1.5.2 Dimension of the support is one

######  Definition 1.18.

The closed support of a permuton ${\boldsymbol{\pi}}$ is the set

$\operatorname{supp}{\boldsymbol{\pi}}:=\mathopen{}\mathclose{{}\left(\text{intersection
of all closed sets $K\subset[0,1]^{2}$ with
${\boldsymbol{\pi}}(K)=1$}}\right).$ (1.4)

For a general permuton, the Hausdorff dimension of its closed support can be
any number in $[1,2]$. For example, the closed support of the uniform measure
on $[0,1]^{2}$ is all of $[0,1]^{2}$. Our next theorem states that the closed
supports of the permutons considered in this paper all have Hausdorff
dimension one.

######  Theorem 1.19.

Assume that we are in the setting of Section 1.2 with an _arbitrary_ choice of
$\gamma\in(0,2)$, $\kappa_{1},\kappa_{2}>4$, and coupling
$(\eta_{1},\eta_{2})$. Let ${\boldsymbol{\pi}}$ be as in (1.2). Let

$\mathcal{T}=\mathcal{T}(\eta_{1},\eta_{2}):=\mathopen{}\mathclose{{}\left\\{(t,s)\in[0,1]^{2}:\eta_{1}(t)=\eta_{2}(s)}\right\\}.$
(1.5)

Almost surely, both the closed support $\operatorname{supp}{\boldsymbol{\pi}}$
and the set $\mathcal{T}$ have Hausdorff dimension one.

It is easy to check that
$\operatorname{supp}{\boldsymbol{\pi}}\subset\mathcal{T}$ (Lemma 2.9), but the
reverse inclusion need not hold. For example, if $\eta_{1}=\eta_{2}$ a.s. then
$\operatorname{supp}{\boldsymbol{\pi}}$ is the diagonal in $[0,1]^{2}$ but
$\mathcal{T}$ contains off-diagonal points which arise from points in
$\mathbbm{C}$ which are hit more than once by $\eta_{1}$ or $\eta_{2}$. It is
in general a subtle question to determine which points of $\mathcal{T}$ belong
to $\operatorname{supp}{\boldsymbol{\pi}}$. We intend to investigate this
further in future work.

The following corollary is a special case of Theorem 1.19. It in particular
answers a question from [Bor21c, Section 1.6].

######  Corollary 1.20.

For each choice of parameters $(\rho,q)\in(-1,1)\times(0,1)$, the closed
support of the skew Brownian permuton with parameters $\rho,q$ has Hausdorff
dimension 1. The same is true for the meandric permuton.

It is shown in [Maa20, Theorem 1.5] that the dimension of the closed support
of the biased Brownian separable permuton (i.e. the skew Brownian permuton
with $\rho=1$ and $q\in(0,1)$; recall Remark 1.10) is equal to 1. In fact, the
one-dimensional Hausdorff measure of its support is bounded above by
$\sqrt{2}$. The proof in [Maa20] is based on a description of the biased
Brownian separable permuton in terms of Brownian motion, so it is possible
(but not obvious) that a related proof could work for the skew Brownian
permutons. However, arguments like those in [Maa20] would not work for the
meandric permuton or for other permutons generated from a pair of independent
space-filling SLE curves.

Our proof of Theorem 1.19, given in Section 4, will be based on a short
argument using estimates for the Liouville quantum gravity metric which come
from [AFS20, GS22].

####  1.5.3 Re-rooting invariance for the meandric permuton

Let $\ell_{n}$ be a uniform meander of size $n$. For
$k\in[1,2n]\cap\mathbbm{Z}$, let $x_{k}$ be the $k$-th intersection point of
$\ell_{n}$ with $\mathbbm{R}$ in left-right order. For
$k\in[1,2n]\cap\mathbbm{Z}$, we obtain a new meander $\ell_{n}^{(k)}$ as
follows. See Figure 5 for an illustration. Let $y\in(-\infty,x_{k})$ be chosen
so that there is no intersection point of $\ell_{n}$ with $\mathbbm{R}$ in the
interval $[y,x_{k})$ and set $f_{y}(z):=1/(z-y)$, for all $z\in\mathbb{C}$. We
then take $\ell_{n}^{(k)}$ to be the meander which is the image of $\ell_{n}$
under the map

  * •

$z\mapsto f_{y}(z)$, if $x_{k}$ is traversed by $\ell_{n}$ from top to bottom;

  * •

$z\mapsto\overline{f_{y}(z)}$, if $x_{k}$ is traversed by $\ell_{n}$ from
bottom to top.

We refer to it as the meander $\ell_{n}$ re-rooted at $x_{k}$. The leftmost
point of $\ell_{n}^{(k)}\cap\mathbbm{R}$ is equal to $1/(x_{k}-y)$. Hence, we
can identify $\ell_{n}^{(k)}$ with the loop $\ell_{n}$, but traversed started
at $x_{k}$ instead of at $x_{1}$.

Figure 5:  Left: A meander $\ell_{n}$ and the corresponding meandric
permutation $\sigma_{n}=1,4,3,2,5,12,7,8,9,10,11,6$. Right: The meander
$\ell_{n}^{(6)}$ obtained by re-rooting $\ell_{n}$ at the point $x_{6}$ and
the corresponding meandric permutation
$\sigma^{(6)}_{n}=1,8,9,10,11,12,7,2,5,4,3,6$.

The mapping $\ell_{n}\rightarrow\ell_{n}^{(k)}$ is a bijection from the set of
meanders of size $n$ to itself. Consequently, $\ell_{n}^{(k)}$ is also a
uniform meander of size $n$. If $\sigma_{n}:=\sigma_{\ell_{n}}$ denotes the
meandric permutation associated with $\ell_{n}$, then the meandric permutation
associated with $\ell_{n}^{(k)}$ is

$\sigma_{n}^{(k)}:=\sigma_{\ell_{n}^{(k)}}=\tau_{\sigma_{n}(k)}^{-1}\circ\sigma_{n}\circ\tau_{k},\quad\text{where}\quad\tau_{j}:=j,j+1,\dots,2n,1,2,\dots,j-1.$
(1.6)

Hence, the law of the uniform meandric permutation is invariant under the
operation of pre- and post- composing with cyclic permutations in such a way
that the resulting permutation still fixes 1. Our next theorem gives an analog
of this property for the meandric permuton. See Figure 6 for an illustration.

Figure 6:  Visual representation of the re-rooting operation of Theorem 1.21.
The re-rooted permuton ${\boldsymbol{\pi}}_{t}$ is obtained by re-arranging
the four rectangles shown in the figure, without changing the restrictions of
the permuton to each of the four rectangles. One can check that conjugating by
cyclic permutations as in (1.6) has a similar effect on the permuton
associated with a given permutation (Definition 1.1).

######  Theorem 1.21.

Suppose that we are in the setting of Section 1.2 with $\gamma\in(0,2)$
arbitrary, $\kappa_{1}=\kappa_{2}=8$, and $(\eta_{1},\eta_{2})$ are
independent modulo time parametrization. Let ${\boldsymbol{\pi}}$ be the
permuton associated with $(\eta_{1},\eta_{2})$ as in (1.2). For $t\in[0,1]$,
let $\psi(t)$ be the (a.s. unique333 The a.s. uniqueness of $\psi(t)$ for a
fixed $t\in[0,1]$ is proven in Lemma 2.10, which also shows that $\psi(t)$ is
the a.s. unique time for which $\eta_{1}(t)=\eta_{2}(\psi(t))$. ) $s\in[0,1]$
for which $(t,s)$ is in the closed support of ${\boldsymbol{\pi}}$. Let
${\boldsymbol{\pi}}_{t}$ be the pushforward of the measure
${\boldsymbol{\pi}}$ under the mapping $[0,1]^{2}\rightarrow[0,1]^{2}$ defined
by

$(u,v)\mapsto\mathopen{}\mathclose{{}\left(u-t-\lfloor
u-t\rfloor,v-\psi(t)-\lfloor v-\psi(t)\rfloor}\right).$

For each fixed $t\in[0,1]$, we have
${\boldsymbol{\pi}}_{t}\overset{d}{=}{\boldsymbol{\pi}}$.

As we will explain in Section 5, Theorem 1.21 is a consequence of a special
re-rooting invariance property for SLE8 (Proposition 5.2). We do not
expect444We plan to give a formal proof of this claim in future work combining
the results in Proposition 5.4 and the aforementioned fact that the field $h$
and the independent SLE curves $\eta_{1}$ and $\eta_{2}$, viewed modulo
conformal and anticonformal automorphisms of $\mathbbm{C}$, are a.s. given by
measurable functions of the associated permuton ${\boldsymbol{\pi}}$ as in
(1.2). that the statement of Theorem 1.21 is true when either $\kappa_{1}$ or
$\kappa_{2}$ is not equal to 8. This suggests that the scaling limit of a
uniform meandric permutation _cannot_ be one of the permutations of Section
1.2 for $\kappa_{1},\kappa_{2}\not=8$, which provides some further evidence in
favor of Conjecture 1.7 (beyond the evidence discussed in Section 6). We do
not have a special property which singles out the LQG parameter
$\gamma=\sqrt{\frac{1}{3}\mathopen{}\mathclose{{}\left(17-\sqrt{145}}\right)}$.

####  1.5.4 Pattern density for the meandric permuton

Let ${\boldsymbol{\pi}}$ be a (possibly random) permuton and $n$ be a positive
integer. Let $(\vec{X},\vec{Y})=(X_{i},Y_{i})_{i\in[1,n]\cap\mathbbm{Z}}$ be
$n$ i.i.d. points in $[0,1]^{2}$ sampled from ${\boldsymbol{\pi}}$. Then the
$n$ points almost surely have distinct $x$ and $y$ coordinates. For
$j\in[1,n]\cap\mathbbm{Z}$, let $\tau(j)$ be such that if $x_{j}$ is the
$j$-th smallest element in $(X_{\ell})_{\ell\in[1,n]\cap\mathbbm{Z}}$, then
$y_{j}$ is the $\tau(j)$-th smallest element in
$(Y_{\ell})_{\ell\in[1,n]\cap\mathbbm{Z}}$. Then $\tau$ is a permutation of
size $n$, which we denote by
$\operatorname{Perm}_{n}({\boldsymbol{\pi}},\vec{X},\vec{Y})$. For a fixed
permutation $\sigma$ of size $n$, the probabilities
$\mathbbm{P}[\operatorname{Perm}_{n}({\boldsymbol{\pi}},\vec{X},\vec{Y})=\sigma]$
and
$\mathbbm{P}[\operatorname{Perm}_{n}({\boldsymbol{\pi}},\vec{X},\vec{Y})=\sigma\,|\,{\boldsymbol{\pi}}]$
are called the annealed and quenched pattern densities of $\sigma$ in
${\boldsymbol{\pi}}$.

If ${\boldsymbol{\pi}}$ is a skew Brownian permuton, it was shown in [BHSY22,
Proposition 1.14] that the annealed pattern densities
$\mathbbm{P}[\operatorname{Perm}_{2}({\boldsymbol{\pi}},\vec{X},\vec{Y})=1,2]$
and
$\mathbb{P}[\operatorname{Perm}_{2}({\boldsymbol{\pi}},\vec{X},\vec{Y})=2,1]$
are $\frac{1}{2}\pm\frac{\theta}{\pi}$, where
$\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ is the angle between the two space-
filling SLE curves $(\eta_{1},\eta_{2})$ in their imaginary geometry coupling
(see Theorem 2.7). The exact relation between $\theta$ and the parameters
$(\rho,q)$ for the skew Brownian permuton is not known. For $n\geq 3$,
although the distribution of
$\operatorname{Perm}_{n}({\boldsymbol{\pi}},\vec{X},\vec{Y})$ can in principle
be expressed by SLE and LQG quantities, the relation would be quite involved;
as explained in [BHSY22, Remark 3.14].

For a class of permutons including the meandric permuton, we have the
following conceptually straightforward expression.

######  Theorem 1.22.

Suppose that we are in the setting of Section 1.2 with $\gamma\in(0,2)$,
$\kappa_{1},\kappa_{2}>4$, and $(\eta_{1},\eta_{2})$ are independent modulo
time parametrization. Recall that $\mu_{h}$ denotes the $\gamma$-LQG area
measure corresponding to a singly marked unit area $\gamma$-Liouville quantum
sphere. Let ${\boldsymbol{\pi}}$ be the permuton associated with
$(\eta_{1},\eta_{2})$ as in (1.2). For $i\in\\{1,2\\}$ and
$z_{1},\cdots,z_{n}\in\mathbb{C}$, let $P_{\kappa_{i}}(z_{1},\cdots,z_{n})$ be
the probability that $\eta_{i}$ hits $z_{j}$ before $z_{j+1}$ for each
$j=1,\dots,n-1$. Let $\zeta(z_{1},\cdots,z_{n})$ be the joint density of $n$
points independently sampled from the measure $\mu_{h}$, i.e.,
$\zeta(z_{1},\cdots,z_{n})$ is the density of the probability measure
$\mathbbm{E}[\mu_{h}\otimes\cdots\otimes\mu_{h}]$ on $\mathbbm{C}^{n}$. For a
permutation $\sigma$ of size $n$, the annealed pattern density is given by

$\mathbbm{P}\mathopen{}\mathclose{{}\left[\operatorname{Perm}_{n}({\boldsymbol{\pi}},\vec{X},\vec{Y})=\sigma}\right]=n!\int_{\mathbbm{C}^{n}}P_{\kappa_{1}}(z_{1},\cdots,z_{n})P_{\kappa_{2}}(z_{\sigma(1)},\cdots,z_{\sigma(n)})\zeta(z_{1},\cdots,z_{n})\prod_{j=1}^{n}dz_{j}.$
(1.7)

###### Proof.

This follows from the independence of $\eta_{1}$ and $\eta_{2}$, and the
definitions of ${\boldsymbol{\pi}}$ and
$\operatorname{Perm}_{n}({\boldsymbol{\pi}},\vec{X},\vec{Y})$. The factor $n!$
enumerates the possible orders in which $\eta_{1}$ can hit $n$ distinct points
on the plane. ∎

The law of the field $h$ in Theorem 1.22 describing a singly marked unit area
$\gamma$-Liouville quantum sphere is not unique (rather, it is only defined up
to a conformal change of coordinates, see Section 2.2). As explained in [BW18,
Section 1.4], for an appropriate choice of $h$ the density function
$\zeta(z_{1},\cdots,z_{n})$ can be expressed as correlation functions in
Liouville conformal field theory on the sphere with all vertex insertions
$\alpha_{1},\dots,\alpha_{n}$ being equal to $\gamma$. These functions were
recently solved exactly via conformal bootstrap; see [GKRV20, Theorem 1.1] and
[GKRV21, Theorem 9.3]. The probability $P_{\kappa}(z_{1},\cdots,z_{n})$ seems
quite difficult to evaluate exactly. We list it as Problem 7.4 in Section 7.

######  Remark 1.23 (Positive quenched pattern density).

For a permutation $\sigma$ of size $n$, let
$\widetilde{\operatorname{occ}}(\sigma,{\boldsymbol{\pi}})=\mathbbm{P}[\operatorname{Perm}_{n}({\boldsymbol{\pi}},\vec{X},\vec{Y})=\sigma\mid{\boldsymbol{\pi}}]$
be the quenched pattern density of $\sigma$ in ${\boldsymbol{\pi}}$. It was
shown in [BHSY22, Theorem 1.10] that
$\widetilde{\operatorname{occ}}(\sigma,{\boldsymbol{\pi}})>0$ a.s. for all
permutations $\sigma$ when ${\boldsymbol{\pi}}$ is a skew Brownian permuton
(with $\rho\neq 1$). Essentially the same arguments as in [BHSY22, Section 4]
show that $\widetilde{\operatorname{occ}}(\sigma,{\boldsymbol{\pi}})>0$ a.s.
in the setting of Theorem 1.22 and also that $P_{\kappa}(z_{1},\dots,z_{n})>0$
for each $\kappa>4$ and each distinct $z_{1},\dots,z_{n}\in\mathbbm{C}$. As
explained in [BHSY22, Corollary 1.8], this implies that random permutations
converging to ${\boldsymbol{\pi}}$ as in Theorem 1.22 have asympotically
positive density for all permutation patterns in the quenched sense.

##  2 Preliminaries

###  2.1 Gaussian free field

In what follows, we use the notation

$B_{r}(z):=\text{open Euclidean ball of radius $r$ centered at $z$}.$ (2.1)

For $z,w\in\mathbbm{C}$, we also define

$|z|_{+}:=\max\\{|z|,1\\}\quad\text{and}\quad
G_{\mathbbm{C}}(z,w):=\log\frac{|z|_{+}|w|_{+}}{|z-w|}.$ (2.2)

The whole-plane Gaussian free field (GFF) is the centered Gaussian random
generalized function (distribution) on $\mathbbm{C}$ with covariances

$\operatorname{Cov}(h^{\mathbbm{C}}(z),h^{\mathbbm{C}}(w)):=G_{\mathbbm{C}}(z,w).$
(2.3)

Note that $h^{\mathbbm{C}}$ is not well-defined pointwise since the covariance
kernel (2.2) diverges to $\infty$ as $z\rightarrow w$. Nevertheless, for
$z\in\mathbbm{C}$ and $r>0$, one can define the average of $h^{\mathbbm{C}}$
over the circle $\partial B_{r}(z)$, which we denote by
$h^{\mathbbm{C}}_{r}(z)$. See [DS11, Section 3.1] for further discussion.

One often views the whole-plane GFF as being defined modulo additive constant.
Our choice of covariance kernel in (2.3) corresponds to fixing the additive
constant so that the circle average $h_{1}^{\mathbbm{C}}(0)$ is zero (see,
e.g., [Var17, Section 2.1.1]). The law of the whole-plane GFF, viewed modulo
additive constant, is invariant under complex affine transformations of
$\mathbbm{C}$. Equivalently,

$h^{\mathbbm{C}}\overset{d}{=}h^{\mathbbm{C}}(a\cdot+b)-h^{\mathbbm{C}}_{|a|}(b),\quad\forall
a\in\mathbbm{C}\setminus\\{0\\},\quad\forall b\in\mathbbm{C}.$ (2.4)

We refer to [She07, BP, WP21] for more background on the GFF.

###  2.2 Liouville quantum gravity

In this subsection we give a brief review of the theory of Liouville quantum
gravity (LQG). Our intention is to provide just enough background for the
reader to understand the proofs in this paper. We refer to [Gwy20, She22] for
brief introduction to LQG and to [BP] for a detailed exposition. Let
$\gamma\in(0,2)$ and let

$Q:=\frac{2}{\gamma}+\frac{\gamma}{2}.$ (2.5)

######  Definition 2.1.

A $\gamma$-Liouville quantum gravity (LQG) surface with $k\in\mathbbm{N}_{0}$
marked points is an equivalence class of $k+2$-tuples
$(U,h,z_{1},\dots,z_{k})$, where

  * •

$U\subset\mathbbm{C}$ is open and $z_{1},\dots,z_{k}\in U\cup\partial U$, with
$\partial U$ viewed as a collection of prime ends;

  * •

$h$ is a generalized function on $U$ (which we will always take to be some
variant of the Gaussian free field);

  * •

$(U,h,z_{1},\dots,z_{k})$ and
$(\widetilde{U},\widetilde{h},\widetilde{z}_{1},\dots,\widetilde{z}_{k})$ are
declared to be equivalent if there is a conformal map
$\phi:\widetilde{U}\rightarrow U$ such that

$\widetilde{h}=h\circ\phi+Q\log|\phi^{\prime}|\quad\text{and}\quad\phi(\widetilde{z}_{j})=z_{j},\quad\forall
j=1,\dots k,$ (2.6)

where $Q$ is as in (2.5).

If $(U,h,z_{1},\dots,z_{k})$ is an equivalence class representative, we refer
to $h$ as an embedding of the quantum surface into $(U,z_{1},\dots,z_{k})$.

In this paper, all of the LQG surfaces we consider will be random, and the
generalized function $h$ will be a GFF plus a continuous function, meaning
that there is a coupling $(h,h^{U})$ of $h$ with the whole-plane or zero-
boundary GFF $h^{U}$ on $U$ (as appropriate) such that a.s. $h-h^{U}$ is a
function which is continuous on $U$ except possibly at finitely many points.

If $h$ is a whole-plane GFF plus a continuous function, one can define the LQG
area measure $\mu_{h}$ on $U$, which is a limit of regularized versions of
$e^{\gamma h}\,d^{2}z$, where $d^{2}z$ denotes Lebesgue measure on $U$ [Kah85,
DS11, RV11]. Almost surely, the measure $\mu_{h}$ assigns positive mass to
every open set and zero mass to every point, but is mutually singular with
respect to Lebesgue measure.

It was shown in [DDDF20, GM21b] that one can also define the LQG metric
$D_{h}$ on $U$, which is a limit of regularized versions of the Riemannian
distance function associated with the Riemannian metric tensor $e^{\gamma
h}(dx^{2}+dy^{2})$, where $dx^{2}+dy^{2}$ is the Euclidean metric tensor on
$U$. Almost surely, the metric $D_{h}$ induces the same topology on $U$ as the
Euclidean metric and is a length metric (i.e., $D_{h}(z,w)$ is the infimum of
the $D_{h}$-lengths of paths in $U$ from $z$ to $w$). The Hausdorff dimension
of $(U,D_{h})$ is a.s. equal to a deterministic, $\gamma$-dependent constant
$d_{\gamma}>2$ which is not known explicitly [GP19, Corollary 1.7]. See
[DDG21] for a survey of known results about the LQG metric.

Both $\mu_{h}$ and $D_{h}$ are compatible with coordinate changes of the form
(2.6). That is, if $\phi:\widetilde{U}\rightarrow U$ is a conformal map and
$h$ and $\widetilde{h}$ are related as in (2.6), then a.s.

$\mu_{h}(A)=\mu_{\widetilde{h}}(\phi^{-1}(A)),\quad\text{$\forall$ Borel set
$A\subset U$}\quad\text{and}\quad
D_{h}(z,w)=D_{\widetilde{h}}(\phi^{-1}(z),\phi^{-1}(w)),\quad\forall z,w\in
U.$ (2.7)

See [DS11, Proposition 2.1] for $\mu_{h}$ and [GM21a, Theorem 1.3] for
$D_{h}$.

Both $\mu_{h}$ and $D_{h}$ are also locally determined by $h$. In the case of
$\mu_{h}$, this means that for each open set $V\subset\mathbbm{C}$, a.s.
$\mu_{h}|_{V}$ is a measurable function of $h|_{V}$ (this follows, e.g., from
the circle average construction of $\mu_{h}$ in [DS11]). In the case of
$D_{h}$, the locality property says the following. For $V\subset\mathbbm{C}$,
we define the internal metric on $V$ to be the metric $D_{h}(\cdot,\cdot;V)$
on $V$ defined by

$D_{h}(z,w;V)=\inf\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\text{$D_{h}$-length
of $P$}}\right)\,:\,\text{$P$ is a path in $V$ from $z$ to $w$}}\right\\}.$
(2.8)

Then a.s. $D_{h}(\cdot,\cdot;V)$ is given by a measurable function of $h|_{V}$
[GM21b, Axiom II].

###  2.3 The quantum sphere

For $\gamma\in(0,2)$, the unit area quantum sphere is the canonical LQG
surface with the topology of the sphere. One sense in which this LQG surface
is canonical is that it is expected, and in some cases proven, to describe the
scaling limit of various types of random planar maps. There are a variety of
different ways of defining the unit area quantum sphere. The first definitions
appeared in [DMS21, DKRV16] and were proven to be equivalent in [AHS17]. The
definition we give here is [AHS17, Definition 2.2] with $k=3$ and
$\alpha_{1}=\alpha_{2}=\alpha_{3}=\gamma$.

######  Definition 2.2.

Using the notation (2.2), we define the Liouville field

$h_{\operatorname{L}}:=h^{\mathbbm{C}}+\gamma G_{\mathbbm{C}}(1,\cdot)+\gamma
G_{\mathbbm{C}}(0,\cdot)-(2Q-\gamma)\log|\cdot|_{+}$ (2.9)

and we let $\mu_{h_{\operatorname{L}}}$ be its associated $\gamma$-LQG
measure.

  * •

The triply marked unit area quantum sphere is the LQG surface
$(\mathbbm{C},h,\infty,0,1)$, where the law $\mathbbm{P}_{h}$ of $h$ is
obtained from the law $\mathbbm{P}_{\tilde{h}_{\operatorname{L}}}$ of

$\tilde{h}_{\operatorname{L}}:=h_{\operatorname{L}}-\frac{1}{\gamma}\log\mu_{h_{\operatorname{L}}}(\mathbbm{C})$
(2.10)

weighted by a $\gamma$-dependent constant times
$\mathopen{}\mathclose{{}\left[\mu_{h_{\operatorname{L}}}(\mathbbm{C})}\right]^{4/\gamma^{2}-2}$,
i.e.
$\frac{d\mathbbm{P}_{h}}{d\mathbbm{P}_{\tilde{h}_{\operatorname{L}}}}=C(\gamma)\cdot\mathopen{}\mathclose{{}\left[\mu_{h_{\operatorname{L}}}(\mathbbm{C})}\right]^{4/\gamma^{2}-2}$.

  * •

The doubly (resp. singly) marked unit area quantum sphere is the doubly (resp.
singly) marked quantum surface obtained from the triply marked unit area
quantum sphere by forgetting the marked point at 1 (resp. the marked points at
1 and 0).

We note that subtracting
$\frac{1}{\gamma}\log\mu_{h_{\operatorname{L}}}(\mathbbm{C})$ in (2.10) makes
it so that the unit area quantum sphere satisfies $\mu_{h}(\mathbbm{C})=1$
a.s. The triply marked quantum sphere has only one possible embedding in
$(\mathbbm{C},\infty,0,1)$ (Definition 2.1). This is because the only
conformal automorphism of the Riemann sphere which fixes $\infty,0,1$ is the
identity. By contrast, the singly (resp. doubly) marked quantum sphere has
multiple possible embeddings, obtained by applying (2.6) when $\phi$ is a
complex affine transformation (resp. a rotation and scaling). Any statements
for a singly or doubly marked quantum sphere are assumed to be independent of
the choice of embedding, unless otherwise specified. Many of the results in
this paper are stated for a singly marked quantum sphere since we only need
one marked point (i.e., the starting point of the SLE curves). However, it is
sometimes convenient to work with a triply marked quantum sphere so that we
can apply the exact description of the field from Definition 2.2.

It is shown in [DMS21, Proposition A.13] that the marked points for the
quantum sphere are independent samples from its associated LQG area measure.

######  Lemma 2.3 ([DMS21]).

Let $(\mathbbm{C},h,\infty,0,1)$ be a triply marked unit area
$\gamma$-Liouville quantum sphere. Conditional on $h$, let
$Z_{1},Z_{2},Z_{3}\in\mathbbm{C}$ be conditionally independent samples from
the $\gamma$-LQG area measure $\mu_{h}$. Then
$(\mathbbm{C},h,Z_{1},Z_{2},Z_{3})$ is a triply marked unit area quantum
sphere, i.e., if
$\phi:\mathbbm{C}\cup\\{\infty\\}\rightarrow\mathbbm{C}\cup\\{\infty\\}$ is a
conformal map such that $\phi(\infty)=Z_{1}$, $\phi(0)=Z_{2}$, and
$\phi(1)=Z_{3}$, then $h\circ\phi+Q\log|\phi^{\prime}|\overset{d}{=}h$.

The following absolute continuity statement allows us to transfer various
results from the whole-plane GFF to the quantum sphere.

######  Lemma 2.4.

Let $(\mathbbm{C},h,\infty,0,1)$ be a triply marked quantum sphere, let
$h^{\mathbbm{C}}$ be a whole-plane GFF, and let

$\widetilde{h}_{*}:=h^{\mathbbm{C}}-\gamma\log|\cdot|-\gamma\log|\cdot-1|\qquad\text{and}\qquad\widetilde{h}:=h^{\mathbbm{C}}-\gamma\log|\cdot|.$
(2.11)

For each $R>0$, the laws of $h|_{B_{R}(0)}$ and
$\widetilde{h}_{*}|_{B_{R}(0)}$, viewed as distributions modulo additive
constant, are mutually absolutely continuous. Furthermore, for each $R>0$, the
laws of $h|_{B_{R}(0)\setminus\overline{B_{1/R}(1)}}$ and
$\widetilde{h}|_{B_{R}(0)\setminus\overline{B_{1/R}(1)}}$, viewed as
distributions modulo additive constant, are mutually absolutely continuous.

###### Proof.

Let $h_{\operatorname{L}}$ be the Liouville field from (2.9), defined using
the same whole-plane GFF $h^{\mathbbm{C}}$ as in (2.11). We first look at the
case of $\widetilde{h}_{*}$. From the definition of $G_{\mathbbm{C}}$ in
(2.2), we see that the difference $f:=\widetilde{h}_{*}-h_{\operatorname{L}}$
is a deterministic continuous function on $\mathbbm{C}$ with locally finite
Dirichlet energy, i.e., $\int_{B_{R}(0)}|\nabla f(z)|^{2}\,d^{2}z<\infty$ for
each $R>0$. From standard local absolute continuity results for the GFF (see,
e.g., [MS17, Proposition 2.9]), the laws of $\widetilde{h}_{*}|_{B_{R}(0)}$
and $h_{\operatorname{L}}|_{B_{R}(0)}$, viewed as distributions modulo
additive constant, are mutually absolutely continuous for each $R>0$. By
Definition 2.2, the laws of $h$ and $h_{\operatorname{L}}$ are mutually
absolutely continuous, viewed as distributions modulo additive constant. This
gives the statement for $\widetilde{h}_{*}$. The statement for $\widetilde{h}$
follows from the statement for $\widetilde{h}_{*}$ since $\gamma\log|\cdot-1|$
is smooth away from 1, so again by [MS17, Proposition 2.9], the fields
$\widetilde{h}|_{B_{R}(0)\setminus\overline{B_{1/R}(1)}}$ and
$\widetilde{h}_{*}|_{B_{R}(0)\setminus\overline{B_{1/R}(1)}}$, viewed as
distributions modulo additive constant, are mutually absolutely continuous. ∎

We now make two remarks which are not needed for the proofs of our main
results, but which are relevant to the physics heuristics in Section 6.

######  Remark 2.5 (Infinite measure on quantum spheres).

There is a natural infinite measure on (free area) quantum spheres which
appears in [DMS21, DKRV16]. To produce a “sample” from the infinite measure on
doubly marked (free area) quantum spheres, we first sample an area $A$ from
the infinite measure $\mathbbm{1}_{(a>0)}a^{-\frac{4}{\gamma^{2}}}\,da$. We
then consider the quantum surface $(\mathbbm{C},h+\frac{1}{\gamma}\log
A,\infty,0)$, where $(\mathbbm{C},h,\infty,0)$ is a doubly marked unit area
quantum sphere [AS21, Lemma 6.12]. The infinite measure on singly marked (free
area) quantum spheres is defined similarly, except that $A$ is instead sampled
from $\mathbbm{1}_{(a>0)}a^{-\frac{4}{\gamma^{2}}-1}\,da$.

######  Remark 2.6 (Physics interpretation and random planar maps connection).

A $\gamma$-LQG surface can be interpreted as a model of gravity (essentially,
random geometry) coupled to matter, where the matter is represented by a
conformal field theory (CFT). The central charge of this CFT is related to the
coupling constant $\gamma\in(0,2)$ by

${\mathbf{c}_{\mathrm{M}}}=25-6\mathopen{}\mathclose{{}\left(\frac{2}{\gamma}+\frac{\gamma}{2}}\right)^{2}\in(-\infty,1).$
(2.12)

This interpretation can be made rigorous by looking at random planar maps.
Indeed, suppose that $(M_{n},S_{n})$ is a random planar map with $n$ edges
decorated by a statistical physics model. We sample $M_{n}$ and $S_{n}$
simultaneously, so the marginal law of $M_{n}$ is given by the uniform measure
on some collection of planar maps weighted by the partition function of
$S_{n}$. Suppose our statistical physics model $S_{n}$ is such that the
scaling limit of the version of $S_{n}$ in a regular lattice (e.g.,
$\mathbbm{Z}^{2}$) is described by a CFT with central charge
${\mathbf{c}_{\mathrm{M}}}$. Then in many cases the scaling limit of $M_{n}$
(e.g., with respect to the Gromov-Hausdorff distance) should be described by
$\gamma$-LQG, where $\gamma$ and ${\mathbf{c}_{\mathrm{M}}}$ are related as in
(2.12). As an example, the scaling limit of the critical Ising model on
$\mathbbm{Z}^{2}$ is described by a CFT with central charge
${\mathbf{c}_{\mathrm{M}}}=1/2$, so random planar maps weighted by the
critical Ising model partition function should converge to LQG with matter
central charge ${\mathbf{c}_{\mathrm{M}}}=1/2$, equivalently,
$\gamma=\sqrt{3}$. See [GHS19, Section 3.1] for more discussion. Using the
formula for the law of the area for the infinite measure on single marked
quantum spheres (Remark 2.5), it is conjectured that the partition function of
$(M_{n},S_{n})$ grows like $A^{n}n^{-\alpha}$ where $A>0$ and

$\alpha=1+\frac{4}{\gamma^{2}}.$ (2.13)

This explains why Conjecture 1.3 is related to the meander exponent
$\alpha=(29+\sqrt{145})/12$ in Section 1.3; see Section 6 for more details.

###  2.4 Space-filling SLE

Let $\kappa>4$. The whole-plane space-filling SLEκ from $\infty$ to $\infty$
is a random non-self-crossing space-filling curve $\eta$ in $\mathbbm{C}$
which starts and ends at $\infty$. We view $\eta$ as a curve defined modulo
monotone increasing time re-parametrization. We give a precise definition of
this curve below. Before we do so, we record some of its qualitative
properties which follow from the construction in [MS17].

  1. $(i)$

When $\kappa\geq 8$, the curve $\eta$ is a two-sided version of chordal SLEκ.
It can be obtained from chordal SLEκ by stopping the chordal SLE curve at the
first time when it hits some fixed interior point $z$, then “zooming in” near
$z$. In particular, for any two times $s<t$ the region $\eta([s,t])$ is simply
connected.

  2. $(ii)$

When $\kappa\in(4,8)$, the curve $\eta$ can be obtained from a two-sided
variant of chordal SLEκ by iteratively “filling in” the bubbles which it
disconnects from its target point. In this case, for typical times $s<t$, the
region $\eta([s,t])$ is not simply connected.

  3. $(iii)$

For each fixed $z\in\mathbbm{C}$, the left and right outer boundaries of
$\eta$ stopped when it hits $z$ are a pair of coupled SLE16/κ-type curves from
$z$ to $\infty$.

  4. $(iv)$

For any times $a<b$, the set $\eta([a,b])$ has non-empty interior. Moreover,
if $\mu$ is a locally finite, non-atomic measure on $\mathbbm{C}$ which
assigns positive mass to every open set, then $\eta$ is continuous if we
parametrize $\eta$ so that $\mu(\eta([a,b]))=b-a$ for each $a<b$.

  5. $(v)$

The law of $\eta$ is invariant under scaling, translation, rotation, and time
reversal, i.e., $a\eta+b$ and $\eta(-\cdot)$ have the same law as $\eta$
modulo time parametrization for each $a\in\mathbbm{C}\setminus\\{0\\}$ and
each $b\in\mathbbm{C}$.

  6. $(vi)$

For each fixed $z\in\mathbbm{C}$, a.s. $\eta$ hits $z$ exactly once. The
maximum number of times that $\eta$ hits any $z\in\mathbbm{C}$ is 3 if
$\kappa\geq 8$ and is a finite, $\kappa$-dependent number if $\kappa\in(4,8)$
[GHM20, Theorem 6.3].

We now give a formal definition of whole-plane space-filling SLEκ, following
[MS17, Section 1.2.3] (see also [DMS21, Section 1.4.1]). The idea of the
definition is based on SLE duality, which says that for $\kappa>4$ the outer
boundary of an SLEκ curve stopped at a given time should consist of one or
more SLE16/κ-type curves [Zha08, Zha10, Dub09, MS16a, MS17]. Let

$\chi:=\frac{\sqrt{\kappa}}{2}-\frac{2}{\sqrt{\kappa}}.$

Let $\widehat{h}$ be a whole-plane GFF viewed modulo additive multiples of
$2\pi\chi$, as in [MS17]. Also fix an angle $\theta\in[-\pi/2,\pi/2]$. For
$z\in\mathbbm{Q}^{2}$, let $\beta_{z}^{L}$ and $\beta_{z}^{R}$ be the flow
lines of $e^{i\widehat{h}/\chi}$ started from $z$, in the sense of [MS17,
Theorem 1.1], with angles $\theta$ and $\theta-\pi$, respectively. Each of
these flow lines is a whole-plane SLE${}_{16/\kappa}(16/\kappa-2)$ curve from
$z$ to $\infty$ (whole-plane SLE${}_{16/\kappa}(\rho)$ for $\rho>-2$ is a
variant of whole-plane SLE introduced in [MS17, Section 2.1]), and the field
$\widehat{h}$ defines a coupling between them. These flow lines will be the
left and right outer boundaries of the space-filling SLEκ $\eta$ stopped when
it hits $z$.

By [MS17, Theorem 1.9], for distinct $z,w\in\mathbbm{Q}^{2}$, the flow lines
$\beta_{z}^{L}$ and $\beta_{z}^{R}$ cannot cross
$\beta_{w}^{L}\cup\beta_{w}^{R}$, but they may hit and bounce off if
$\kappa\in(4,8)$. Furthermore, a.s. the flow lines $\beta_{z}^{L}$ and
$\beta_{w}^{L}$ (resp. $\beta_{z}^{R}$ and $\beta_{w}^{R}$) eventually merge
into each other. See Figure 7 for an illustration.

Figure 7:  The flow lines $\beta_{z}^{L}$ (in red) and $\beta_{z}^{R}$ (in
blue) for several points $z\in\mathbbm{C}$. For each $z$, these flow lines are
the left and right outer boundaries, respectively, of $\eta$ stopped when it
hits $z$. The figure shows the case when $\kappa\geq 8$. When
$\kappa\in(4,8)$, the flow lines started from the same point can hit each
other and bounce off (c.f. Figure 10). The space-filling SLEκ fills in the
regions 1,2,3,4,5,6, and 7 in this order.

We define a total ordering on $\mathbbm{Q}^{2}$ by declaring that $z$ comes
before $w$ if and only if $w$ lies in a connected component of
$\mathbbm{C}\setminus(\beta_{z}^{L}\cup\beta_{z}^{R})$ which lies to the right
of $\beta_{z}^{L}$ and to the left of $\beta_{z}^{R}$. It follows from [MS17,
Theorem 1.16] (see also [DMS21, Footnote 4]) that there is a unique space-
filling path $\eta$ from $\infty$ to $\infty$ which hits the points of
$\mathbbm{Q}^{2}$ in the prescribed order and is continuous when it is
parametrized, e.g., so that it traverses one unit of (two-dimensional)
Lebesgue measure in one unit of time. This curve $\eta$ is defined to be the
space-filling SLEκ counterflow line of $\widehat{h}$ from $\infty$ to $\infty$
of angle $\pi/2-\theta$.

The law of the curve $\eta$ does not depend on $\theta$, but the coupling of
$\eta$ with $\widehat{h}$ depends on $\theta$. In particular, if
$\theta\in(-\pi/2,\pi/2)$ and $\eta_{1}$ and $\eta_{2}$ are the space-filling
SLE counterflow lines of $\widehat{h}$ with angles 0 and $\pi/2-\theta$,
respectively, then $\eta_{1}$ and $\eta_{2}$ are coupled together in a non-
trivial way. By [MS17, Theorem 1.16], each of $\eta_{1}$ and $\eta_{2}$ is
a.s. given by a measurable function of the other. The reason for our interest
in this coupling is the following result, which is [Bor21c, Theorem 1.17] and
which is proven building on [DMS21, GHS16].

######  Theorem 2.7 ([Bor21c]).

Fix $\rho\in(-1,1)$ and $q\in[0,1]$. Also let $\gamma\in(0,2)$ and $\kappa>4$
satisfy $\rho=-\cos(\pi\gamma^{2}/4)$ and $\kappa=16/\gamma^{2}$. There exists
$\theta=\theta(\rho,q)\in[-\pi/2,\pi/2]$ such that the following is true. Let
$(\mathbbm{C},h,\infty)$ be a singly marked unit area $\gamma$-Liouville
quantum gravity sphere. Also let $\eta_{1},\eta_{2}$ be the space-filling SLEκ
counterflow lines of angles 0 and $\pi/2-\theta$ of a common whole-plane GFF
as above, sampled independently from $h$ and then parametrized so that they
each traverse one unit of $\gamma$-LQG mass in one unit of time. Then the
permutation ${\boldsymbol{\pi}}$ associated with $(\eta_{1},\eta_{2})$ as in
(1.2) coincides with the skew Brownian permuton with parameters $\rho$ and
$q$.

The exact correspondence between the parameters $\theta$ and $q$ is not known,
but it is known that $\theta(\rho,1/2)=\pi/2$ and that
$q\mapsto\theta(\rho,q)$ is a homeomorphism from $[0,1]$ to $[-\pi/2,\pi/2]$
[Bor21c, Remark 1.18].

###  2.5 Basic properties of permutons constructed from SLEs and LQG

Assume that we are in the setting of Section 1.2. In particular,
$(\mathbbm{C},h,\infty)$ is a singly marked unit area $\gamma$-Liouville
quantum sphere and $(\eta_{1},\eta_{2})$ is a coupling of a space-filling
SLE${}_{\kappa_{1}}$ curve and a space-filling SLE${}_{\kappa_{2}}$ curve,
sampled independently from $h$ and then parametrized by $\mu_{h}$-mass;
$\psi:[0,1]\rightarrow[0,1]$ satisfies $\eta_{1}(t)=\eta_{2}(\psi(t))$; and
${\boldsymbol{\pi}}$ is the permuton as in (1.2). In this subsection, we will
establish some basic properties of ${\boldsymbol{\pi}}$. We first check that
${\boldsymbol{\pi}}$ is well-defined, and give an equivalent definition.

######  Lemma 2.8.

Almost surely, the permuton ${\boldsymbol{\pi}}$ is well-defined, i.e., the
definition does not depend on the choice of $\psi$. Moreover, a.s. for each
rectangle $[a,b]\times[c,d]\subset[0,1]^{2}$,

${\boldsymbol{\pi}}\mathopen{}\mathclose{{}\left([a,b]\times[c,d]}\right)=\mu_{h}\mathopen{}\mathclose{{}\left(\eta_{1}([a,b])\cap\eta_{2}([c,d])}\right).$
(2.14)

We note that a similar statement to Lemma 2.8 is proven for the skew Brownian
permuton in [BHSY22, Proposition 1.13].

###### Proof of Lemma 2.8.

The formula (2.14) uniquely determines ${\boldsymbol{\pi}}$ and does not
depend on $\psi$, so it suffices to prove (2.14). For each $z\in\mathbbm{C}$,
a.s. $z$ is not a multiple point of $\eta_{2}$, i.e., $\eta_{2}$ hits $z$
exactly once. Since $h$ is independent from $(\eta_{1},\eta_{2})$ and
$\eta_{1}$ and $\eta_{2}$ are parametrized by $\mu_{h}$-mass, a.s. the set of
times $t\in[0,1]$ such that $\eta_{1}(t)$ is a multiple point of $\eta_{2}$
has zero Lebesgue measure.

If $\eta_{1}(t)$ is not a multiple point of $\eta_{2}$, then for any choice of
$\psi$ satisfying (1.1), $\psi(t)\in[c,d]$ if and only if
$\eta_{1}(t)\in\eta_{2}([c,d])$. By the preceding paragraph, a.s. this is the
case for Lebesgue-a.e. $t\in[0,1]$, so a.s. for every rectangle
$[a,b]\times[c,d]$,

$\displaystyle{\boldsymbol{\pi}}\mathopen{}\mathclose{{}\left([a,b]\times[c,d]}\right)$
$\displaystyle=\operatorname{Leb}\mathopen{}\mathclose{{}\left\\{t\in[a,b]:\psi(t)\in[c,d]}\right\\}\quad\text{(by~{}\eqref{eqn-
permuton-def})}$
$\displaystyle=\operatorname{Leb}\mathopen{}\mathclose{{}\left\\{t\in[a,b]:\eta_{1}(t)\in\eta_{2}([c,d])}\right\\}$
$\displaystyle=\mu_{h}\mathopen{}\mathclose{{}\left(\eta_{1}([a,b])\cap\eta_{2}([c,d])}\right),$

where the last equality is because $\eta_{1}$ is parametrized by
$\mu_{h}$-mass. ∎

We also have the following basic lemma about the closed support of
${\boldsymbol{\pi}}$ (Definition 1.18).

######  Lemma 2.9.

Almost surely, for any choice of the function $\psi$ from (1.1),

$\operatorname{supp}{\boldsymbol{\pi}}\subset\overline{\\{(t,\psi(t)):t\in[0,1]\\}}\subset\\{(t,s)\in[0,1]:\eta_{1}(t)=\eta_{2}(s)\\}=\mathcal{T}.$
(2.15)

###### Proof.

The first inclusion in (2.15) is immediate from the definition (1.2). The
second inclusion follows from the fact that $\eta_{1}(t)=\eta_{2}(\psi(t))$
for each $t\in[0,1]$ by definition; and the fact that the set
$\\{(t,s)\in[0,1]:\eta_{1}(s)=\eta_{1}(t)\\}$ is closed (by the continuity of
$\eta_{1}$ and $\eta_{2}$). ∎

Both inclusions in (2.15) can potentially be strict. For example, if
$\eta_{1}=\eta_{2}$ then $\operatorname{supp}{\boldsymbol{\pi}}$ is the
diagonal in $[0,1]^{2}$. The third set in (2.15) includes off-diagonal points,
which arise from pairs of distinct times $(t,s)$ such that
$\eta_{1}(t)=\eta_{1}(s)$ or $\eta_{2}(t)=\eta_{2}(s)$ (which are multiple
points of $\eta_{1}$ or $\eta_{2}$). The middle set in (2.15) can include off-
diagonal points or not, depending on the choice of $\psi$.

The following lemma tells us that the ambiguity in the choice of $\psi$ can
arise only from multiple points of $\eta_{2}$.

######  Lemma 2.10.

Almost surely, for any choice of the function $\psi$ from (1.1), the following
is true. For each time $t\in[0,1]$ such that $\eta_{1}(t)$ is hit only once by
$\eta_{2}$, we have that $\psi(t)$ is the unique $s\in[0,1]$ such that
$(t,s)\in\operatorname{supp}{\boldsymbol{\pi}}$. Furthermore, for each fixed
$t\in[0,1]$, a.s. $\psi(t)$ is the unique $s\in[0,1]$ such that
$(t,s)\in\operatorname{supp}{\boldsymbol{\pi}}$.

###### Proof.

Since ${\boldsymbol{\pi}}$ is a permuton, we have
${\boldsymbol{\pi}}([a,b]\times[0,1])=b-a$ for each $0\leq a\leq b\leq 1$.
From this, we infer that $\operatorname{supp}{\boldsymbol{\pi}}$ intersects
$[a,b]\times[0,1]$ for each $0\leq a<b\leq 1$. Since
$\operatorname{supp}{\boldsymbol{\pi}}$ is closed, this implies that
$\operatorname{supp}{\boldsymbol{\pi}}$ intersects $\\{t\\}\times[0,1]$ for
each $t\in[0,1]$. By Lemma 2.9, if $\eta_{2}$ hits $\eta_{1}(t)$ only once,
there is only one possible intersection point of
$\operatorname{supp}{\boldsymbol{\pi}}$ with $\\{t\\}\times[0,1]$, namely
$(t,\psi(t))$. This gives the first assertion of the lemma. To deduce the last
statement, we note that if $t\in[0,1]$ is fixed, then $\eta_{1}(t)$ is
independent from $\eta_{2}$ (viewed modulo time parametrization), and hence
a.s. $\eta_{2}$ hits $\eta_{1}(t)$ only once (recall property (vi) at the
beginning of Section 2.4). ∎

##  3 Longest increasing subsequence is sublinear

In this section we will prove Theorem 1.12, which asserts that the longest
increasing subsequence is sublinear for permutations which converge to
${\boldsymbol{\pi}}$.

###  3.1 Longest increasing subsequences and monotone sets for permutons

To prove Theorem 1.12, we will use a general criterion on a permuton which
implies a sublinear bound on the longest increasing subsequence for converging
permutations, which we now state and prove.

Figure 8: A set $A\subset[0,1]^{2}$ is monotone if for any point $(t,s)\in A$,
the set $A$ is contained in the union of the green rectangles in the figure.

######  Definition 3.1.

We say that a set $A\subset[0,1]^{2}$ is monotone if

$A\subset\mathopen{}\mathclose{{}\left([0,t]\times[0,s]}\right)\cup\mathopen{}\mathclose{{}\left([t,1]\times[s,1]}\right),\quad\forall(t,s)\in
A.$ (3.1)

See Figure 8 for an illustration. An example of a monotone set is the graph of
a non-decreasing function $[0,1]\rightarrow[0,1]$, or a subset of such a
graph. The reason for our interest in monotone sets is the following
deterministic result.

######  Proposition 3.2.

Let ${\boldsymbol{\pi}}$ be a permuton and let
$\\{\sigma_{n}\\}_{n\in\mathbbm{N}}$ be a sequence of permutations of size
$|\sigma_{n}|\rightarrow\infty$ whose associated permutons
${\boldsymbol{\pi}}_{\sigma_{n}}$ converge weakly to ${\boldsymbol{\pi}}$.
Assume that ${\boldsymbol{\pi}}(A)=0$ for every monotone set
$A\subset[0,1]^{2}$ (Definition 3.1). Then the longest increasing subsequence
of $\sigma_{n}$ is sublinear, i.e.,
$\operatorname{LIS}(\sigma_{n})/|\sigma_{n}|\rightarrow 0$.

###### Proof.

We will prove the contrapositive, i.e., we will assume that the largest
increasing subsequence of $\sigma_{n}$ is not sublinear and show that there is
a monotone subset of $[0,1]^{2}$ with positive ${\boldsymbol{\pi}}$-mass.

Our assumption implies that after possibly replacing
$\\{\sigma_{n}\\}_{n\in\mathbbm{N}}$ by a subsequence, we can find $c>0$ such
that for each $n\in\mathbbm{N}$, there is an increasing subsequence of
$\sigma_{n}$ of length at least $c|\sigma_{n}|$. That is, for each
$n\in\mathbbm{N}$, there is a set
$L_{n}\subset[1,|\sigma_{n}|]\cap\mathbbm{Z}$ such that $\\#L_{n}\geq
c|\sigma_{n}|$ and $\sigma_{n}|_{L_{n}}$ is monotone increasing. For
$n\in\mathbbm{N}$, let

$A_{n}:=\bigcup_{j\in
L_{n}}\mathopen{}\mathclose{{}\left[\frac{j-1}{|\sigma_{n}|},\frac{j}{|\sigma_{n}|}}\right]\times\mathopen{}\mathclose{{}\left[\frac{\sigma_{n}(j)-1}{|\sigma_{n}|},\frac{\sigma_{n}(j)}{|\sigma_{n}|}}\right]\subset[0,1]^{2}.$
(3.2)

By Definition 1.1 of ${\boldsymbol{\pi}}_{\sigma_{n}}$, we have
${\boldsymbol{\pi}}_{\sigma_{n}}(A_{n})\geq c$.

By compactness, after possibly passing to a further subsequence of
$\\{\sigma_{n}\\}_{n\in\mathbbm{N}}$, we can arrange that $A_{n}$ converges in
the Hausdorff distance to a closed set $A\subset[0,1]^{2}$. We first argue
that ${\boldsymbol{\pi}}(A)>0$. Indeed, for each $\varepsilon>0$, it holds for
each large enough $n\in\mathbbm{N}$ that $A_{n}$ is contained in the Euclidean
$\varepsilon$-neighborhood $B_{\varepsilon}(A)$. Since
${\boldsymbol{\pi}}_{\sigma_{n}}\rightarrow{\boldsymbol{\pi}}$ weakly,

${\boldsymbol{\pi}}(\overline{B_{\varepsilon}(A)})\geq\liminf_{n\rightarrow\infty}{\boldsymbol{\pi}}_{\sigma_{n}}(\overline{B_{\varepsilon}(A)})\geq\liminf_{n\rightarrow\infty}{\boldsymbol{\pi}}_{\sigma_{n}}(A_{n})\geq
c.$ (3.3)

Sending $\varepsilon\rightarrow 0$ gives ${\boldsymbol{\pi}}(A)\geq c>0$.

We next claim that $A$ is a monotone set in the sense of Definition 3.1. To
this end, let $(t,s)\in A$. Since $A_{n}\rightarrow A$ in the Hausdorff
distance, by (3.2) we can find a $j_{n}\in L_{n}$ such that
$(j_{n},\sigma_{n}(j_{n}))/|\sigma_{n}|\rightarrow(t,s)$ as
$n\rightarrow\infty$. Since $\sigma_{n}$ is monotone on $L_{n}$, it holds for
each $n\in\mathbbm{N}$ that

$A_{n}\subset\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,\frac{j_{n}}{|\sigma_{n}|}}\right]\times\mathopen{}\mathclose{{}\left[0,\frac{\sigma_{n}(j_{n})}{|\sigma_{n}|}}\right]}\right)\cup\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[\frac{j_{n}}{|\sigma_{n}|},1}\right]\times\mathopen{}\mathclose{{}\left[\frac{\sigma_{n}(j_{n})}{|\sigma_{n}|},1}\right]}\right).$
(3.4)

Since $(j_{n},\sigma_{n}(j_{n}))/|\sigma_{n}|\rightarrow(t,s)$, for each
$\varepsilon>0$ it holds for each large enough $n\in\mathbbm{N}$ that the set
on the right side of (3.4) is contained in the $\varepsilon$-neighborhood of
$([0,t]\times[0,s])\cup([t,1]\times[s,1])$. Since $A_{n}\rightarrow A$, it
follows that for each $\varepsilon>0$,

$A\subset\mathopen{}\mathclose{{}\left(\text{$\varepsilon$-neighborhood of
$([0,t]\times[0,s])\cup([t,1]\times[s,1])$}}\right).$ (3.5)

Sending $\varepsilon\rightarrow 0$ shows that $A$ is a monotone set. ∎

###  3.2 Monotone sets for permutons constructed from SLEs and LQG

Henceforth assume that we are in the setting of Theorem 1.12. To prove the
theorem, we need to check the condition of Proposition 3.2 for the random
permuton ${\boldsymbol{\pi}}$. This turns out to be straightforward using SLE
and LQG theory.

Let $\eta_{1}$ and $\eta_{2}$ be the coupled or independent space-filling SLE
curves of parameters $\kappa_{1}$ and $\kappa_{2}$ from Theorem 1.12. For
$z\in\mathbbm{C}$, let $\mathcal{Q}_{z}$ be the set of $w\in\mathbbm{C}$ such
that $\eta_{1}$ hits $w$ before it hits $z$ and $\eta_{2}$ hits $w$ after it
hits $z$. More precisely, $w\in\mathcal{Q}_{z}$ if and only if

$\exists t_{1},s_{1},t_{2},s_{2}\in[0,1]\quad\text{such that}\quad t_{1}\leq
s_{1},\quad t_{2}\leq
s_{2},\quad\eta_{1}(s_{1})=\eta_{2}(t_{2})=z,\quad\eta_{1}(t_{1})=\eta_{2}(s_{2})=w.$
(3.6)

Note that $\mathcal{Q}_{z}$ is a closed set and that $\mathcal{Q}_{z}$ depends
only on $\eta_{1}$ and $\eta_{2}$ viewed modulo time parametrization, so
$\mathcal{Q}_{z}$ is independent from $h$. The definition of $\mathcal{Q}_{z}$
is slightly simpler if $z$ is hit only once by each of $\eta_{1}$ or
$\eta_{2}$ (which is true for $\mu_{h}$-a.e. $z\in\mathbbm{C}$), since in this
case there is only one possible choice for each of $s_{1}$ and $t_{2}$. See
Figure 9 for an illustration of $\mathcal{Q}_{z}$.

To prove Theorem 1.12, we will first show that for $\mu_{h}$-a.e.
$z\in\mathbbm{C}$, the set $\mathcal{Q}_{z}$ has positive density with respect
to $\mu_{h}$, i.e., we will prove the following lemma.

Figure 9: Illustration of the set $\mathcal{Q}_{z}$ of (3.6) for three of the
different pairs of SLE curves $(\eta_{1},\eta_{2})$ considered in Theorem
1.12. Here, $z$ is a point which is hit only once by each of $\eta_{1}$ and
$\eta_{2}$. In each case, $\mathcal{Q}_{z}$ is the light blue region,
$\eta_{1}$ travels from south to north and $\eta_{2}$ travels from west to
east. The left and right boundaries of $\eta_{1}$ (resp. $\eta_{2}$) stopped
when it hits $z$ are shown in red and blue (resp. green and purple). The set
$\mathcal{Q}_{z}$ is the region lying to the south of the union of the red and
blue curves and to the east of the union of the green and purple curves. In
the case of the skew Brownian permuton, $\eta_{1}$ and $\eta_{2}$ are two
space-filling counterflow lines of a GFF with angles $0$ and $\pi/2-\theta$
(Theorem 2.7). This case is illustrated in the left and middle panels. The
red, blue, green, and purple curves do not cross each other, but may hit each
other and bounce off (as shown in the second panel). One can determine the
values of $\kappa$ and $\theta$ for which this happens using [MS17,
Proposition 3.28] and [Dub09, Lemma 15]. In the case of two independent space-
filling SLEs (right panel), the red and blue curves can cross the green and
purple curves, so the region $\mathcal{Q}_{z}$ is not connected. The three
pictures in the case when $\kappa_{1}\in(4,8)$ (resp. $\kappa_{2}\in(4,8)$)
are similar, except that the red and blue curves (resp. the green and purple
curves) can hit each other. Our proof is the same for all of the different
cases considered in Theorem 1.12.

######  Lemma 3.3.

Assume that we are in the setting of Theorem 1.12. Almost surely,

$\limsup_{r\rightarrow
0}\frac{\mu_{h}\mathopen{}\mathclose{{}\left(\mathcal{Q}_{z}\cap
B_{r}(z)}\right)}{\mu_{h}(B_{r}(z))}>0,\quad\text{for $\mu_{h}$-a.e.\
$z\in\mathbbm{C}$},$ (3.7)

where $B_{r}(z)$ is a Euclidean ball as in (2.1).

We will eventually show that every monotone set for ${\boldsymbol{\pi}}$ has
zero $\mu_{h}$-mass by combining Lemma 3.3 with the Lebesgue density theorem
for the Radon measure $\mu_{h}$ (see the proof of Proposition 3.6). We will
prove Lemma 3.3 via a scaling argument. To do this, we need to work in an
infinite-volume setting, i.e., we need to work with a field which satisfies an
exact spatial scale invariance property in law (the unit area quantum sphere
does not satisfy such a property). We choose to work with a whole-plane GFF
with a $\gamma$-log singularity at 0 since this field has the same local
behavior near 0 as a triply marked quantum sphere (Lemma 2.4).

######  Lemma 3.4.

Let $\gamma\in(0,2)$ and $\kappa_{1},\kappa_{2}>4$. Let $\widetilde{h}$ be a
whole-plane GFF plus $\gamma\log|\cdot|^{-1}$. Let $\eta_{1}$ and $\eta_{2}$
be two whole-plane space-filling SLE curve from $\infty$ to $\infty$ of
parameters $\kappa_{1}$ and $\kappa_{2}$, sampled independently from
$\widetilde{h}$. Assume that $(\eta_{1},\eta_{2})$ are coupled together in one
of the two possible ways listed just before Theorem 1.12. Define the set
$\mathcal{Q}_{0}$ as in (3.6). Almost surely,

$\limsup_{r\rightarrow
0}\frac{\mu_{\widetilde{h}}\mathopen{}\mathclose{{}\left(\mathcal{Q}_{0}\cap
B_{r}(0)}\right)}{\mu_{\widetilde{h}}(B_{r}(0))}>0.$ (3.8)

###### Proof.

We first argue that a.s. $\mathcal{Q}_{0}$ has non-empty interior. For
$i\in\\{1,2\\}$, let $\tau_{0}^{i}$ be the (a.s. unique) time when $\eta_{i}$
hits 0. By the construction of space-filling SLE in Section 2.4, the set
$\partial\eta_{i}((-\infty,\tau_{0}^{i}])$ is the union of two
SLE${}_{16/\kappa_{i}}$-type curves $\beta_{i}^{L}$ and $\beta_{i}^{R}$ from 0
to $\infty$, namely, the left and right boundaries of
$\eta_{i}((-\infty,\tau_{0}^{i}])$. The set $\mathcal{Q}_{0}$ is the closure
of the intersection of the region which lies to the left of $\beta_{1}^{L}$
and to the right of $\beta_{1}^{R}$; and the region which lies to the right of
$\beta_{2}^{L}$ and to the left of $\beta_{2}^{R}$. See Figure 9 for an
illustration. Due to our choice of coupling $(\eta_{1},\eta_{2})$, the
intersection of any two of the four curves
$\beta_{1}^{L},\beta_{1}^{R},\beta_{2}^{L},\beta_{2}^{R}$ is a totally
disconnected set.555 If $\eta_{1}$ and $\eta_{2}$ are independent modulo time
parametrization, this follows from the fact that two independent SLE16/κ-type
curves a.s. do not trace each other for a non-trivial interval of time, which
in turn follows from the fact that an SLE16/κ-type curve a.s. does not spend a
positive amount of time tracing a fixed deterministic curve with zero Lebesgue
measure. If $(\eta_{1},\eta_{2})$ are coupled together as in Theorem 2.7, the
statement instead follows from interaction rules for flow lines of a whole-
plane Gaussian free field with different angles [MS17, Proposition 3.28].
Since each of these four curves is simple being an SLE of parameter
$16/\kappa_{i}<4$, topological considerations now show that $\mathcal{Q}_{0}$
has non-empty interior.

The joint law of $(\eta_{1},\eta_{2})$, viewed as curves modulo time
parametrization, is invariant under scaling space. Hence the same is true of
the law of $\mathcal{Q}_{0}$. Therefore, a.s. $\mathcal{Q}_{0}\cap B_{r}(0)$
has non-empty interior for every $r>0$. Since a.s. $\mu_{\widetilde{h}}$
assigns positive mass to every open set, we get that a.s.
$\mu_{\widetilde{h}}(\mathcal{Q}_{0}\cap B_{r}(0))>0$ for every $r>0$.

By the LQG coordinate change formula (2.7), a.s. for each $r>0$,

$\mu_{\widetilde{h}(r\cdot)+Q\log r}(A)=\mu_{\widetilde{h}}(rA),\quad\text{for
each Borel set $A\subset\mathbbm{C}$.}$ (3.9)

Furthermore, by (2.4), the law of $\widetilde{h}$ is scale invariant modulo
additive constant, i.e., for each $r>0$ there is a random $C_{r}>0$ such that
$\widetilde{h}(r\cdot)-C_{r}\overset{d}{=}\widetilde{h}$. Adding a constant to
$\widetilde{h}$ results in scaling $\mu_{\widetilde{h}}$ by a constant factor,
so does not change the ratios of the $\mu_{\widetilde{h}}$-masses of any two
sets. From this, (3.9), and the scale invariance of the law of
$\mathcal{Q}_{0}$, we get that the law of
$\mu_{\widetilde{h}}\mathopen{}\mathclose{{}\left(\mathcal{Q}_{0}\cap
B_{r}(0)}\right)/\mu_{\widetilde{h}}(B_{r}(0))$ does not depend on $r$. As we
noted above, this random variable is a.s. positive.

Consequently, for each $p\in(0,1)$ there exists $c>0$ such that

$\mathbbm{P}\mathopen{}\mathclose{{}\left[\frac{\mu_{\widetilde{h}}\mathopen{}\mathclose{{}\left(\mathcal{Q}_{0}\cap
B_{r}(0)}\right)}{\mu_{\widetilde{h}}(B_{r}(0))}\geq c}\right]\geq
p,\quad\forall r>0.$ (3.10)

Hence, with probability at least $p$, there exist arbitrarily small values of
$r>0$ for which

$\frac{\mu_{\widetilde{h}}\mathopen{}\mathclose{{}\left(\mathcal{Q}_{0}\cap
B_{r}(0)}\right)}{\mu_{\widetilde{h}}(B_{r}(0))}\geq c.$

Since $p$ is arbitrary, the limsup in (3.8) is a.s. positive. ∎

###### Proof of Lemma 3.3.

Conditional on $(h,\eta_{1},\eta_{2})$, let $Z,W$ be sampled independently
from the probability measure $\mu_{h}$. It suffices to show that a.s.

$\limsup_{r\rightarrow
0}\frac{\mu_{h}\mathopen{}\mathclose{{}\left(\mathcal{Q}_{Z}\cap
B_{r}(Z)}\right)}{\mu_{h}(B_{r}(Z))}>0.$ (3.11)

By Lemma 2.3, the law of $(\mathbbm{C},h,\infty,Z,W)$ is that of a triply
marked unit area quantum sphere. That is, if
$\phi:\mathbbm{C}\rightarrow\mathbbm{C}$ is the complex affine map with
$\phi(0)=Z$ and $\phi(1)=W$, then the surface
$(\mathbbm{C},h\circ\phi+Q\log|\phi^{\prime}|,\infty,0,1)$ has the law
described in Definition 2.2. By Lemma 2.4, the restriction of
$h\circ\phi+Q\log|\phi^{\prime}|$ to $B_{1/2}(0)$ is absolutely continuous
with respect to the corresponding restriction of a whole-plane GFF minus
$\gamma\log|\cdot|$, with both random distributions viewed modulo additive
constant. We note that adding a constant to the field does not change any of
the ratios in (3.11).

The law of $(\eta_{1},\eta_{2})$, viewed modulo time parametrization, is
invariant under scaling and spatial translation, and $(\eta_{1},\eta_{2})$ is
independent from $(Z,W)$, hence from $\phi$. Consequently,
$\phi^{-1}(\mathcal{Q}_{Z})\overset{d}{=}\mathcal{Q}_{0}$. From this, the
absolute continuity in the preceding paragraph, and Lemma 3.4, we get that
a.s.

$\limsup_{r\rightarrow
0}\frac{\mu_{h\circ\phi+Q\log|\phi^{\prime}|}\mathopen{}\mathclose{{}\left(\phi^{-1}(\mathcal{Q}_{Z})\cap
B_{r}(0)}\right)}{\mu_{h\circ\phi+Q\log|\phi^{\prime}|}(B_{r}(0))}>0.$ (3.12)

By the LQG coordinate change formula (2.7) and the fact that $\phi$ is complex
affine, so $\phi(B_{r}(0))=B_{|\phi^{\prime}(0)|r}(Z)$, we see that (3.12)
implies (3.11). ∎

We now prove an SLE version of the “monotone sets have zero mass” condition
from Proposition 3.2.

######  Proposition 3.5.

Assume that we are in the setting of Theorem 1.12. Almost surely, the
following is true. Let $X\subset\mathbbm{C}$ be a $\mu_{h}$-measurable set
with the following property: for $\mu_{h}$-a.e. pair of points $z,w\in X$, the
curves $\eta_{1}$ and $\eta_{2}$ hit $z$ and $w$ in the same order. Then
$\mu_{h}(X)=0$.

###### Proof.

For $z\in\mathbbm{C}$, let $\mathcal{Q}_{z}$ be as in (3.6). By Lemma 3.3,
a.s.

$\limsup_{r\rightarrow
0}\frac{\mu_{h}\mathopen{}\mathclose{{}\left(\mathcal{Q}_{z}\cap
B_{r}(z)}\right)}{\mu_{h}(B_{r}(z))}>0,\quad\text{for $\mu_{h}$-a.e.\
$z\in\mathbbm{C}$}.$ (3.13)

By the Lebesgue density theorem for general Radon measures on $\mathbbm{C}$
[Mat95, Corollary 2.14], a.s. for every $\mu_{h}$-measurable set
$X\subset\mathbbm{C}$,

$\lim_{r\rightarrow 0}\frac{\mu_{h}(B_{r}(z)\cap
X)}{\mu_{h}(B_{r}(z))}=1,\quad\text{for $\mu_{h}$-a.e.\ $z\in X$}.$ (3.14)

If $z$ and $w$ are two points which are hit in the same order by $\eta_{1}$
and $\eta_{2}$, and $z$ and $w$ are each hit at most once by each of
$\eta_{1}$ and $\eta_{2}$, then the definition (3.6) of $\mathcal{Q}_{z}$
shows that $w\notin\mathcal{Q}_{z}$. From this observation and the fact that
a.s. $\mu_{h}$-a.e. $z\in\mathbbm{C}$ is hit at most once by each of
$\eta_{1}$ and $\eta_{2}$, we see that if $X$ is a $\mu_{h}$-measurable set
with the property in the proposition statement, then
$\mu_{h}(X\cap\mathcal{Q}_{z})=0$ for $\mu_{h}$-a.e. $z\in X$. By combining
this with (3.13), we obtain that a.s., for every such set $X$,

$\limsup_{r\rightarrow 0}\frac{\mu_{h}(B_{r}(z)\cap
X)}{\mu_{h}(B_{r}(z))}<1,\quad\text{for $\mu_{h}$-a.e.\ $z\in X$}.$ (3.15)

By (3.14) and (3.15), we get that $\mu_{h}(X)=0$. ∎

######  Proposition 3.6.

Assume that we are in the setting of Theorem 1.12. Almost surely, for each
monotone set $A\subset[0,1]^{2}$ (Definition 3.1), we have
${\boldsymbol{\pi}}(A)=0$.

###### Proof.

Let $\psi:[0,1]\rightarrow[0,1]$ be a function as in the definition (1.2) of
${\boldsymbol{\pi}}$, so that $\eta_{1}(t)=\eta_{2}(\psi(t))$ for each
$t\in[0,1]$. For a monotone set $A\subset[0,1]^{2}$, let

$T_{A}=\mathopen{}\mathclose{{}\left\\{t\in[0,1]:(t,\psi(t))\in A}\right\\}.$
(3.16)

We claim that a.s. for every monotone set $A$, it holds that $\mu_{h}$-a.e.
pair of points $z,w\in\eta_{1}(T_{A})$ are hit in the same order by $\eta_{1}$
and $\eta_{2}$. Almost surely, the set of points in $\mathbbm{C}$ which are
hit more than once by either $\eta_{1}$ or $\eta_{2}$ has zero $\mu_{h}$-mass,
so we can assume that each of $z$ and $w$ is hit exactly once by each of
$\eta_{1}$ and $\eta_{2}$. Fix $z,w\in T_{A}$ and let $t_{z}\in T_{A}$ and
$t_{w}\in T_{A}$ be the times when $\eta_{1}$ hits $z$ and $w$, respectively,
and assume without loss of generality that $t_{z}<t_{w}$. By definition of
$T_{A}$, $(t_{z},\psi(t_{z}))$ and $(t_{w},\psi(t_{w}))$ belong to $A$ which
is monotone. By Definition 3.1 of a monotone set and since $t_{z}<t_{w}$, it
follows that $\psi(t_{z})<\psi(t_{w})$. Since $\eta_{2}$ hits each of $z$ and
$w$ only once, $\psi(t_{z})$ and $\psi(t_{w})$ are the unique times at which
$\eta_{2}$ hits $z$ and $w$. Hence $z$ and $w$ are hit in the same order by
$\eta_{1}$ and $\eta_{2}$.

By Proposition 3.5 and the preceding claim, it is a.s. the case that for every
monotone set $A$, the $\mu_{h}$-mass of $\eta_{1}(T_{A})$ is zero. Since
$\eta_{1}$ is parametrized by $\mu_{h}$-mass, this implies that the one-
dimensional Lebesgue measure of $T_{A}$ is zero. By the definition (1.2) of
${\boldsymbol{\pi}}$, this means that ${\boldsymbol{\pi}}(A)=0$. ∎

###### Proof of Theorem 1.12.

The statement for the longest increasing subsequence follows combining
Propositions 3.2 and 3.6. The statement for the longest decreasing subsequence
can be obtained by the same argument. Alternatively, one can observe that if
$\eta_{1}$ and $\eta_{2}$ are a pair of SLE curves as in Theorem 1.12, then
the pair consisting of $\eta_{1}$ and the time reversal of $\eta_{2}$ also
satisfies the conditions of Theorem 1.12. ∎

##  4 Dimension of the support is one

In this section we will prove Theorem 1.19, which asserts that the dimension
of the closed support $\operatorname{supp}{\boldsymbol{\pi}}$ is one for _any_
of the permutons defined in Section 1.2. We recall from Theorem 1.19 that
$\mathcal{T}=\\{(t,s)\in[0,1]^{2}:\eta_{1}(t)=\eta_{2}(s)\\}$ .

Since ${\boldsymbol{\pi}}$ is a permuton, by definition it is immediate that
the projection of $\operatorname{supp}{\boldsymbol{\pi}}$ onto the first
coordinate of $[0,1]^{2}$ is all of $[0,1]$. Since the projection map is
1-Lipschitz, this implies that the Hausdorff dimension of
$\operatorname{supp}{\boldsymbol{\pi}}$ is at least one. From this and Lemma
2.9, we get that a.s.

$1\leq\dim_{\mathcal{H}}\operatorname{supp}{\boldsymbol{\pi}}\leq\dim_{\mathcal{H}}\mathcal{T},$
(4.1)

where $\dim_{\mathcal{H}}$ denotes the Hausdorff dimension of a set. It
therefore suffices to show that a.s. $\dim_{\mathcal{H}}\mathcal{T}\leq 1$.

###  4.1 Core argument

By the definition of $\mathcal{T}$, if $[a,b]\times[c,d]\subset[0,1]^{2}$ is a
rectangle, then $\mathcal{T}$ intersects $[a,b]\times[c,d]$ if and only if
$\eta_{1}([a,b])\cap\eta_{2}([c,d])\not=\emptyset$. We therefore seek an upper
bound for the probability that the images of two small time intervals under
$\eta_{1}$ and $\eta_{2}$ intersect. The following lemma is the main input in
the proof of Theorem 1.19.

######  Lemma 4.1.

Assume that we are in the setting of Theorem 1.19. Fix $\zeta\in(0,1)$ and
$\delta\in(0,1/2)$. There exist
$E_{\varepsilon}=E_{\varepsilon}(\zeta,\delta)\in\sigma(h,\eta_{1},\eta_{2})$
for $\varepsilon>0$ such that $\lim_{\varepsilon\rightarrow
0}\mathbbm{P}[E_{\varepsilon}]=1$ and the following is true. Let
$S\in[\delta,1-\delta]$ be sampled uniformly from Lebesgue measure on
$[\delta,1-\delta]$, independently from everything else. For each
$t\in[\delta,1-\delta]$ and each small enough $\varepsilon>0$ (depending on
$\delta$),

$\mathbbm{P}\mathopen{}\mathclose{{}\left[E_{\varepsilon},\,\eta_{1}([t,t+\varepsilon])\cap\eta_{2}([S,S+\varepsilon])\not=\emptyset}\right]\leq\varepsilon^{1-\zeta}.$
(4.2)

When we apply Lemma 4.1, we will take $\zeta$ and $\delta$ to be small. Note
that the statement of the lemma would not be true if we took $S$ to be a fixed
deterministic time instead of a random time. The reason is that the coupling
of $\eta_{1}$ and $\eta_{2}$, viewed as curves modulo time parametrization, is
arbitrary. For example, we allow $\eta_{1}=\eta_{2}$ in which case the
probability that
$\eta_{1}([t,t+\varepsilon])\cap\eta_{2}([t,t+\varepsilon])\not=\emptyset$ is
1.

The idea of the proof of Lemma 4.1 is as follows. Since $\eta_{1}$ and
$\eta_{2}$ are parametrized by LQG mass, the set $\eta([t,t+\varepsilon])$ has
$\mu_{h}$-mass $\varepsilon$ and the conditional law of $\eta_{2}(S)$ given
$(h,\eta_{1},\eta_{2})$ is that of a sample from the restriction of
$(1-2\delta)^{-1}\mu_{h}$ to $\eta_{2}([\delta,1-\delta])$. Therefore, the
probability that $\eta_{2}(S)\in\eta_{1}([t,t+\varepsilon])$ is at most of
order $\varepsilon$. To deduce an upper bound for the probability in (4.2), we
will improve this estimate by showing that it is unlikely for $\eta_{2}(S)$ to
be “close” to $\eta_{1}([t,t+\varepsilon])$ and that it is unlikely for
$\eta_{2}([S,S+\varepsilon])$ to be “large”.

It is convenient to measure “close” and “large” in terms of the $\gamma$-LQG
metric $D_{h}$ associated with $h$. This is because a space-filling SLE curve
parametrized by LQG mass behaves nicely with respect to this metric (see Lemma
4.2 below). In what follows, we let $d_{\gamma}$ be the Hausdorff dimension of
$\mathbbm{C}$ with respect to the metric $D_{h}$, which is a.s. equal to a
finite deterministic constant [GP19, Corollary 1.7]. We will need the
following two lemmas.

######  Lemma 4.2.

Let $\gamma\in(0,2)$ and $\kappa>4$. Let $(\mathbbm{C},h,\infty)$ be a singly
marked unit area $\gamma$-Liouville quantum sphere. Let
$\eta:[0,1]\rightarrow\mathbbm{C}$ be a whole-plane space-filling SLEκ from
$\infty$ to $\infty$ sampled independently from $h$, then parametrized by
$\gamma$-LQG mass with respect to $h$. Also fix $\zeta\in(0,1)$ and
$\delta\in(0,1/2)$. Almost surely, for each small enough $\varepsilon>0$ (how
small is random),

$D_{h}(\eta(s),\eta(t))\leq\varepsilon^{1/d_{\gamma}-\zeta},\quad\forall
s,t\in[\delta,1-\delta]\quad\text{such that}\quad|s-t|\leq\varepsilon.$ (4.3)

For the next lemma, we introduce the notation

$\mathcal{B}_{r}(z;D_{h}):=\mathopen{}\mathclose{{}\left(\text{$D_{h}$-ball of
radius $r$ centered at $z$}}\right),\quad\forall
z\in\mathbbm{C}\cup\\{\infty\\},\quad\forall r>0.$ (4.4)

######  Lemma 4.3.

Let $(\mathbbm{C},h,\infty)$ be a singly marked unit area $\gamma$-Liouville
quantum sphere. Also fix $\zeta\in(0,1)$. Almost surely, for each small enough
$r>0$ (how small is random),

$r^{d_{\gamma}+\zeta}\leq\mu_{h}\mathopen{}\mathclose{{}\left(\mathcal{B}_{r}(z;D_{h})}\right)\leq
r^{d_{\gamma}-\zeta},\quad\forall z\in\mathbbm{C}\cup\\{\infty\\}.$ (4.5)

Lemmas 4.2 and 4.3 are straightforward consequences of known estimates for the
LQG metric [AFS20, GS22], but some work is needed to transfer from the case of
a whole-plane GFF (which is the case treated by the results we cite) to the
case of a unit area quantum sphere. We postpone the proofs until after the
proof of Theorem 1.19.

###### Proof of Lemma 4.1.

Fix $\widetilde{\zeta}\in(0,\zeta)$ to be chosen later, in a deterministic
manner depending only on $\gamma$ and $\zeta$. Let
$E_{\varepsilon}=E_{\varepsilon}(\widetilde{\zeta},\delta)$ be the event that
the following is true:

  1. $(i)$

For each $i\in\\{1,2\\}$, we have
$D_{h}(\eta_{i}(s),\eta_{i}(t))\leq\varepsilon^{1/d_{\gamma}-\widetilde{\zeta}}$
for each $s,t\in[\delta/2,1-\delta/2]$ such that $|s-t|\leq\varepsilon$;

  2. $(ii)$

$\mu_{h}\mathopen{}\mathclose{{}\left(\mathcal{B}_{r}(z;D_{h})}\right)\leq
r^{d_{\gamma}-\widetilde{\zeta}}$ for each
$r\in(0,2\varepsilon^{1/d_{\gamma}-\widetilde{\zeta}}]$.

By Lemma 4.2 (applied once to each of $\eta_{1}$ and $\eta_{2}$) and Lemma
4.3, we have $\mathbbm{P}[E_{\varepsilon}]\rightarrow 1$ as
$\varepsilon\rightarrow 0$.

We henceforth assume that $\varepsilon<\delta/2$. Let $t$ and $S$ be as in the
lemma statement. If $E_{\varepsilon}$ occurs, then by condition (i) in the
definition of $E_{\varepsilon}$,

$\eta_{1}([t,t+\varepsilon])\subset\mathcal{B}_{\varepsilon^{1/d_{\gamma}-\widetilde{\zeta}}}(\eta_{1}(t);D_{h})\quad\text{and}\quad\eta_{2}([S,S+\varepsilon])\subset\mathcal{B}_{\varepsilon^{1/d_{\gamma}-\widetilde{\zeta}}}(\eta_{2}(S);D_{h}).$
(4.6)

By condition (ii) in the definition of $E_{\varepsilon}$, on this event the
$\mu_{h}$-mass of the larger ball
$\mathcal{B}_{2\varepsilon^{1/d_{\gamma}-\widetilde{\zeta}}}(\eta_{1}(t);D_{h})$
is at most a $(\gamma,\widetilde{\zeta})$-dependent constant times
$\varepsilon^{(d_{\gamma}-\widetilde{\zeta})(1/d_{\gamma}-\widetilde{\zeta})}$.
Since $S$ is sampled from Lebesgue measure on $[\delta,1-\delta]$,
independently from everything else, and $\eta_{2}$ is parametrized by
$\mu_{h}$-mass, the conditional law of $\eta_{2}(S)$ given
$(h,\eta_{1},\eta_{2})$ is given by the restriction of
$(1-2\delta)^{-1}\mu_{h}$ to $\eta_{2}([\delta,1-\delta])$. Therefore,

$\mathbbm{P}\mathopen{}\mathclose{{}\left[E_{\varepsilon},\,\eta_{2}(S)\in\mathcal{B}_{2\varepsilon^{1/d_{\gamma}-\widetilde{\zeta}}}(\eta_{1}(t);D_{h})}\right]\preceq\mathbbm{E}\mathopen{}\mathclose{{}\left[\mathbbm{1}_{E_{\varepsilon}}\mu_{h}\mathopen{}\mathclose{{}\left(\mathcal{B}_{2\varepsilon^{1/d_{\gamma}-\widetilde{\zeta}}}(\eta_{1}(t);D_{h})}\right)}\right]\preceq\varepsilon^{(d_{\gamma}-\widetilde{\zeta})(1/d_{\gamma}-\widetilde{\zeta})},$
(4.7)

where $\preceq$ denotes inequality up to a finite constant which depends only
on $\gamma,\widetilde{\zeta},\delta$.

By (4.6) and the triangle inequality for $D_{h}$, if $E_{\varepsilon}$ occurs
and
$\eta_{2}(S)\notin\mathcal{B}_{2\varepsilon^{1/d_{\gamma}-\widetilde{\zeta}}}(\eta_{1}(t);D_{h})$,
then $\eta_{1}([t,t+\varepsilon])\cap\eta_{2}([S,S+\varepsilon])=\emptyset$.
Therefore, (4.7) implies that

$\mathbbm{P}\mathopen{}\mathclose{{}\left[E_{\varepsilon},\,\eta_{1}([t,t+\varepsilon])\cap\eta_{2}([S,S+\varepsilon])\not=\emptyset}\right]\preceq\varepsilon^{(d_{\gamma}-\widetilde{\zeta})(1/d_{\gamma}-\widetilde{\zeta})}.$

We now obtain (4.2) by choosing $\widetilde{\zeta}>0$ to be small enough that
$(d_{\gamma}-\widetilde{\zeta})(1/d_{\gamma}-\widetilde{\zeta})<1-\zeta$. ∎

###### Proof of Theorem 1.19.

By (4.1), we only need to show that a.s. $\dim_{\mathcal{H}}\mathcal{T}\leq
1$. Fix $\zeta\in(0,1)$ and $\delta\in(0,1/2)$. By the countable stability of
Hausdorff dimension, it suffices to show that a.s.

$\dim_{\mathcal{H}}\mathopen{}\mathclose{{}\left(\mathcal{T}\cap[\delta,1-\delta]^{2}}\right)\leq
1+\zeta.$ (4.8)

To this end, let $E_{\varepsilon}$ be the event of Lemma 4.1 with $\zeta/2$ in
place of $\zeta$, and recall that $\mathbbm{P}[E_{\varepsilon}]\rightarrow 1$
as $\varepsilon\rightarrow 0$. We will upper-bound the expected number of
$\varepsilon\times\varepsilon$ squares needed to cover the set in (4.8) on the
event $E_{\varepsilon}$.

Since $\mathcal{T}=\\{(t,s)\in[0,1]^{2}:\eta_{1}(t)=\eta_{2}(s)\\}$, for
$x,y\in[0,1-\varepsilon]^{2}$, the set $\mathcal{T}$ intersects
$[x,x+\varepsilon]\times[y,y+\varepsilon]$ if and only if
$\eta_{1}([x,x+\varepsilon])\cap\eta_{2}([y,y+\varepsilon])\not=\emptyset$.
Therefore, the expected number of $\varepsilon\times\varepsilon$ squares
needed to cover $\mathcal{T}\cap[\delta,1-\delta]^{2}$, truncated on the event
$E_{\varepsilon}$, is at most

$\mathbbm{E}\mathopen{}\mathclose{{}\left[\mathbbm{1}_{E_{\varepsilon}}\sum_{x\in[\delta-\varepsilon,1-\delta]\cap\varepsilon\mathbbm{Z}}\sum_{y\in[\delta-\varepsilon,1-\delta]\cap\varepsilon\mathbbm{Z}}\mathbbm{1}_{\eta_{1}([x,x+\varepsilon])\cap\eta_{2}([y,y+\varepsilon])\not=\emptyset}}\right].$
(4.9)

For each $x,y\in[\delta-\varepsilon,1-\delta]\cap\varepsilon\mathbbm{Z}$, the
square $[x,x+\varepsilon]\times[y,y+\varepsilon]$ is contained in
$[t,t+2\varepsilon]\times[s,s+2\varepsilon]$ for each $t\in[x-\varepsilon,x]$
and each $s\in[y-\varepsilon,y]$. Therefore, the quantity (4.9) is bounded
above by

$\displaystyle\varepsilon^{-2}\mathbbm{E}\mathopen{}\mathclose{{}\left[\mathbbm{1}_{E_{\varepsilon}}\int_{\delta-2\varepsilon}^{1-\delta}\int_{\delta-2\varepsilon}^{1-\delta}\mathbbm{1}_{\eta_{1}([t,t+2\varepsilon])\cap\eta_{2}([s,s+2\varepsilon])\not=\emptyset}\,ds\,dt}\right]$
$\displaystyle\qquad\qquad=\varepsilon^{-2}\mathbbm{P}\mathopen{}\mathclose{{}\left[E_{\varepsilon},\,\eta_{1}([T,T+2\varepsilon])\cap\eta_{2}([S,S+2\varepsilon])\not=\emptyset}\right],$
(4.10)

where $S$ and $T$ are sampled uniformly and independently from
$[\delta-2\varepsilon,1-\delta]$, independently from everything else.

By Lemma 4.1 (with $\zeta/2$ in place of $\zeta$), for each small enough
$\varepsilon>0$ the right side of (4.1) is at most $\varepsilon^{-1-\zeta/2}$.
Hence the expected number of $\varepsilon\times\varepsilon$ squares needed to
cover $\mathcal{T}\cap[\delta,1-\delta]^{2}$ on the event $E_{\varepsilon}$ is
at most $\varepsilon^{-1-\zeta/2}$. We obtain (4.8) by combining this with
Markov’s inequality and the fact that $\mathbbm{P}[E_{\varepsilon}]\rightarrow
1$. ∎

###  4.2 Proofs of LQG metric estimates

To conclude the proof of Theorem 1.19, it remains to prove our estimates for
the LQG metric, namely Lemmas 4.2 and 4.3. We will prove these estimates by
applying known results for the whole-plane GFF, then using local absolute
continuity, in the form of Lemma 2.4, to transfer these results to a singly
marked unit-area $\gamma$-Liouville quantum sphere $(\mathbbm{C},h,\infty)$.

By the LQG coordinate change formula (2.7) for $\mu_{h}$ and for $D_{h}$, the
statements of both lemmas do not depend on the particular choice of embedding
$h$ (recall Definition 2.1). Hence, we can assume without loss of generality
that our embedding is chosen in such a way that $(\mathbbm{C},h,\infty,0,1)$
is a triply marked unit area quantum sphere. Due to Lemma 2.3, such an
embedding can be obtained by sampling points $Z,W\in\mathbbm{C}$ from
$\mu_{h}$, then applying a complex affine map which takes $Z$ to 0 and $W$ to
1.

Let

$\widetilde{h}=h^{\mathbbm{C}}-\gamma\log|\cdot|,$ (4.11)

where $h^{\mathbbm{C}}$ is a whole-plane GFF normalized so that its average
over the unit circle is zero. For $R>0$, we define the open set

$U_{R}:=B_{R}(0)\setminus\overline{B_{1/R}(1)}.$ (4.12)

By Lemma 2.4,

For each $R>0$, the laws of $h|_{U_{R}}$ and $\widetilde{h}|_{U_{R}}$, viewed
modulo additive constant, are mutually absolutely continuous. (4.13)

###### Proof of Lemma 4.2.

Step 1: estimate for the whole-plane GFF. Fix $\zeta\in(0,1)$. With
$\widetilde{h}$ as in (4.11), let $\widetilde{\eta}$ be a whole-plane space-
filling SLEκ from $\infty$ to $\infty$ sampled independently from
$\widetilde{h}$ and then parametrized by $\gamma$-LQG mass with respect to
$\widetilde{h}$. It is shown in [GS22, Theorem 1.4] that666 Strictly speaking,
[GS22, Theorem 1.4] considers the circle average embedding $h^{\gamma}$ of a
$\gamma$-quantum cone instead of the field $\widetilde{h}$ of (4.11). However,
the statement for $\widetilde{h}$ can easily be deduced from the statement for
$h^{\gamma}$ using the fact that the restrictions of $\widetilde{h}$ and
$h^{\gamma}$ to the unit disk agree in law (see [DMS21, Definition 4.10] and
the discussion just after) along with the scaling property of the whole-plane
GFF (2.4).  an analog of the lemma statement holds with $\widetilde{h}$ in
place of $h$. Namely, a.s. for each $R>0$ it holds for each small enough
$\varepsilon>0$ (how small is random and depends on $R$ and $\zeta$) that

$D_{\widetilde{h}}(\widetilde{\eta}(s),\widetilde{\eta}(t))\leq\varepsilon^{1/d_{\gamma}-\zeta},\quad\forall
s,t\in\mathbbm{R}\quad\text{such
that}\quad|s-t|\leq\varepsilon\quad\text{and}\quad\widetilde{\eta}(s),\widetilde{\eta}(t)\in
B_{2R}(0).$ (4.14)

Step 2: absolute continuity. We will now deduce the analogue result as in
(4.14) when $\widetilde{h}$ is replaced with $h$ using an absolute continuity
argument. Recall the internal metric $D_{\widetilde{h}}(\cdot,\cdot;V)$ for
$V\subset\mathbbm{C}$ open which is defined in (2.8). As explained just after
(2.8), this metric is a measurable function of $\widetilde{h}|_{V}$.

If $\varepsilon>0$ is small enough that
$\varepsilon^{1/d_{\gamma}-\zeta}<D_{\widetilde{h}}\mathopen{}\mathclose{{}\left(U_{R},\partial
U_{2R}}\right)$, then $D_{\widetilde{h}}(z,w)=D_{\widetilde{h}}(z,w;U_{2R})$
for each $z,w\in U_{R}$ with $|z-w|\leq\varepsilon^{1/d_{\gamma}-\zeta}$. The
locality property of $D_{\widetilde{h}}$, as explained in the preceding
paragraph, therefore implies that for such a choice of $\varepsilon$, the
restricted distance function

$\mathopen{}\mathclose{{}\left(D_{\widetilde{h}}(z,w):z,w\in
U_{R},\>D_{\widetilde{h}}(z,w)\leq\varepsilon^{1/d_{\gamma}-\zeta}}\right)$

is a.s. determined by $\widetilde{h}|_{U_{2R}}$. Therefore, for small enough
$\varepsilon>0$, the event that (4.14) holds is determined by
$\widetilde{h}|_{U_{2R}}$ and $\widetilde{\eta}$, viewed modulo time
parametrization.

By (4.2), there are random $C,\widetilde{C}\in\mathbbm{R}$ (depending on $R$
but not on $\varepsilon$) such that the laws of $(h+C)|_{U_{2R}}$ and
$(\widetilde{h}+\widetilde{C})|_{U_{2R}}$ are mutually absolutely continuous.
Adding a constant to $\widetilde{h}$ scales LQG areas and distances by a
constant factor. Since $\eta$ and $\widetilde{\eta}$ agree in law modulo time
parametrization, combining these facts with the conclusion of the preceding
paragraph shows that (4.14) also holds for $h$, i.e., a.s. for each $R>0$ it
holds for each small enough $\varepsilon>0$ that

$D_{h}(\eta(s),\eta(t))\leq\varepsilon^{1/d_{\gamma}-\zeta},\quad\forall
s,t\in[0,1]\quad\text{such
that}\quad|s-t|\leq\varepsilon\quad\text{and}\quad\eta(s),\eta(t)\in U_{R}.$
(4.15)

Step 3: transferring to $\eta([\delta,1-\delta])$. Let $T$ be the (a.s.
unique) time at which $\eta$ hits 1. Almost surely,
$\eta([T-\delta,T+\delta])$ contains a neighborhood of 1 and
$\eta([\delta,1-\delta])$ is a compact subset of $\mathbbm{C}$. Therefore,
a.s. there exists $R>0$ such that

$\eta\mathopen{}\mathclose{{}\left([\delta,1-\delta]\setminus[T-\delta,T+\delta]}\right)\subset
U_{R}.$

From this and (4.15), we obtain the following restricted version of the lemma
statement. Almost surely, for each small enough $\varepsilon>0$,

$D_{h}(\eta(s),\eta(t))\leq\varepsilon^{1/d_{\gamma}-\zeta},\quad\forall
s,t\in[\delta,1-\delta]\setminus[T-\delta,T+\delta]\quad\text{such
that}\quad|s-t|\leq\varepsilon.$ (4.16)

To remove the restriction that $s,t\notin[T-\delta,T+\delta]$, we re-sample
the marked point at 1. Let $S$ be sampled uniformly from $[0,1]$,
independently from everything else. Since $\eta$ is parametrized by
$\mu_{h}$-mass, the conditional law of $\eta(S)$ given $(h,\eta)$ is that of a
sample from $\mu_{h}$. By Lemma 2.3, the triply marked quantum surface
$(\mathbbm{C},h,\infty,0,\eta(S))$ is a triply marked unit area quantum
sphere. Hence, if $\phi:\mathbbm{C}\rightarrow\mathbbm{C}$ is the complex
affine map which fixes zero and satisfies $\phi(1)=\eta(S)$, then
$(h\circ\phi+Q\log|\phi^{\prime}|,\phi^{-1}\circ\eta)\overset{d}{=}(h,\eta)$.
By (4.16) with $(h\circ\phi+Q\log|\phi^{\prime}|,\phi^{-1}\circ\eta)$ instead
of $(h,\eta)$ together with the LQG coordinate change formula for $\mu_{h}$
and $D_{h}$ (2.7), we get that (4.16) also holds with $S$ instead of $T$.
Since $S$ is uniformly distributed on $[0,1]$ and is independent from $h$ and
$\eta$, the version of (4.16) with $S$ instead of $T$ implies the lemma
statement. ∎

######  Remark 4.4.

At the end of the proof of Lemma 4.2, we could have instead argued that the
conditional law of $T$ given the curve-decorated quantum surface
$(\mathbbm{C},h,\infty,0,\eta)$ is uniform on $[0,1]$, but we found it easier
to explain the argument by introducing $S$.

###### Proof of Lemma 4.3.

By [AFS20, Theorem 1.1],777The statement of [AFS20, Theorem 1.1] is given for
a whole-plane GFF without a log singularity. However, the proof still works
for a whole-plane GFF plus $\gamma\log|\cdot|^{-1}$. Alternatively, the
statement for a whole-plane GFF plus $\gamma\log|\cdot|^{-1}$ can be deduced
by applying the statement for a whole-plane GFF in logarithmically many dyadic
annuli centered at the origin and taking a union bound. we have the following
analog of the lemma statement with the field $\widetilde{h}$ from (4.11) in
place of $h$. For each $R>0$, it is a.s. the case that for each small enough
$r>0$,

$r^{d_{\gamma}+\zeta}\leq\mu_{\widetilde{h}}\mathopen{}\mathclose{{}\left(\mathcal{B}_{r}(z;D_{h})}\right)\leq
r^{d_{\gamma}-\zeta},\quad\forall z\in B_{R}(0).$ (4.17)

By a similar absolute continuity argument as in the proof of Lemma 4.2, we
deduce from (4.17) that for each $R>0$, it is a.s. the case that for each
small enough $r>0$, the estimate (4.17) holds with $h$ in place of
$\widetilde{h}$.

The law of the triply marked quantum sphere $(\mathbbm{C},h,\infty,0,1)$ is
invariant under interchanging the marked points at 0 and $\infty$ (this
follows, e.g., from Lemma 2.3). Hence, $h(1/\cdot)\overset{d}{=}h$. We can
therefore deduce from (4.17) with $h$ in place of $\widetilde{h}$ that for
each $R>0$, a.s. for each small enough $r>0$,

$r^{d_{\gamma}+\zeta}\leq\mu_{h}\mathopen{}\mathclose{{}\left(\mathcal{B}_{r}(z;D_{h})}\right)\leq
r^{d_{\gamma}-\zeta},\quad\forall z\in\mathbbm{C}\setminus B_{1/R}(0).$ (4.18)

Combining (4.17) (with $h$ in place of $\widetilde{h}$) and (4.18) gives the
lemma statement. ∎

##  5 Re-rooting invariance

In this section we will prove the re-rooting invariance property for permutons
constructed from $\gamma$-LQG with $\gamma\in(0,2)$ decorated by two
independent SLE8 curves, as given by Theorem 1.21.

###  5.1 Re-rooting invariance for SLE8

The main input in the proof of Theorem 1.21 is a re-rooting property for SLE8.
To state this property, we make the following definition.

######  Definition 5.1 (Re-rooting a curve).

Let $P:[0,1]\rightarrow\mathbbm{C}\cup\\{\infty\\}$ be a curve in the Riemann
sphere with $P(0)=P(1)$ and $P([0,1])=\mathbbm{C}\cup\\{\infty\\}$. For
$z\in\mathbbm{C}$, let $\tau_{z}$ be the first time at which $P$ hits $z$. We
define the re-rooting of $P$ at $z$ to be the curve
$P^{z}:[0,1]\rightarrow\mathbbm{C}\cup\\{\infty\\}$ obtained by concatenating
$P|_{[\tau_{z},1]}$ followed by $P|_{[0,\tau_{z}]}$. That is,

$P^{z}(t):=\begin{cases}P(t+\tau_{z}),\quad&\text{if $t\in[0,1-\tau_{z}]$},\\\
P(t-(1-\tau_{z})),\quad&\text{if $t\in[1-\tau_{z},1]$}.\end{cases}$ (5.1)

If we think of $P$ as a space-filling loop based at the point $P(0)=P(1)$,
then $P^{z}$ is the same loop with the base point taken to be $z$ instead. The
following lemma tells us that whole-plane space-filling SLEκ is re-rooting
invariant if and only if $\kappa=8$.

######  Proposition 5.2 (Re-rooting invariance for SLE8).

Let $\kappa>4$ and let $\eta$ be a whole-plane space-filling SLEκ from
$\infty$ to $\infty$. Assume $\eta$ is parametrized by the interval $[0,1]$,
so that $\eta:[0,1]\rightarrow\mathbbm{C}\cup\\{\infty\\}$. If $\kappa=8$,
then for any $z\in\mathbbm{C}$ the re-rooted curve $\eta^{z}$ has the law of a
whole-plane space-filling SLEκ from $z$ to $z$, viewed modulo time
parametrization. Equivalently, if $\phi^{z}$ is a fractional linear
transformation taking $z$ to $\infty$ (i.e., $\phi^{z}(w)=a/(z-w)$ for some
non-zero $a\in\mathbbm{C}$), then $\phi^{z}\circ\eta^{z}$ has the same law as
$\eta$, viewed modulo time parametrization. If $\kappa\not=8$, then this
property is not true for any $z\in\mathbbm{C}$.

As our proof will show, the reason why the symmetry property of Proposition
5.2 holds only for $\kappa=8$ is that chordal space-filling SLEκ is reversible
if and only if $\kappa=8$ [MS17, Theorem 1.19].

###### Proof of Proposition 5.2.

By the construction of space-filling SLE from flow lines of the whole-plane
Gaussian free field (Section 2.4), together with the translation invariance of
the law of the whole-plane GFF, viewed modulo additive constant (2.4), the law
of $\eta$, viewed modulo time parametrization, is invariant under spatial
translation. We can therefore assume without loss of generality that $z=0$.

In [MS17, Section 1.2.3] the authors construct a chordal version of space-
filling SLEκ between two marked boundary points of a simply connected domain
via a similar construction to the one described in Section 2.4. As explained
in [DMS21, Footnote 4], one can construct the whole-plane space-filling SLEκ
from the chordal version as follows. See Figure 10 for an illustration.

  * •

Sample a pair of whole-plane SLE${}_{16/\kappa}(2-16/\kappa)$ curves
$\beta^{L},\beta^{R}$ from 0 to $\infty$, coupled together so that they can be
realized as the flow lines of angle $\pi/2$ and $-\pi/2$ for the whole-plane
GFF. Equivalently, first sample a whole-plane SLE${}_{16/\kappa}(2-16/\kappa)$
curve $\beta^{L}$ from 0 to $\infty$; then, conditional on $\beta^{L}$, sample
a chordal SLE${}_{16/\kappa}(-8/\kappa;-8/\kappa)$ from 0 to $\infty$ in the
complement of the first curve.888 Whole-plane SLE${}_{16/\kappa}(\rho)$ and
chordal SLE${}_{\kappa}(\rho^{L};\rho^{R})$ for $\rho,\rho^{L},\rho^{R}>-2$
are variants of SLE16/κ. We will not need the precise definitions of these
processes here. We refer to [MS17, Section 2.2] and [MS16a, Section 2.2],
respectively, for more background on these processes.

  * •

The curves $\beta^{L}$ and $\beta^{R}$ divide space into two open domains:
$U_{-}$, which lies to the left and $\beta^{L}$ and to the right of
$\beta^{R}$, and $U_{+}$ which lies to the right of $\beta^{L}$ and to the
left of $\beta^{R}$.

  * •

If $\kappa\geq 8$, since $\beta^{L}$ and $\beta^{R}$ do not hit each other
except at their starting point, the regions $U_{-}$ and $U_{+}$ are simply
connected. Conditional on $\beta^{L}$ and $\beta^{R}$, let $\eta_{-}$ (resp.
$\eta_{+}$) be a chordal SLEκ from 0 to $\infty$ in $U_{-}$ (resp. $U_{+}$).
Let $\eta$ be the concatenation of the time reversal of $\eta_{-}$ followed by
$\eta_{+}$.

  * •

If $\kappa\in(4,8)$, then $\beta^{L}$ and $\beta^{R}$ intersect in an
uncountable totally disconnected set which is hit in the same order by
$\beta^{L}$ and $\beta^{R}$. In this case, each of $U_{-}$ and $U_{+}$
consists of a countable ordered string of connected components (“beads”), each
of which has one boundary arc which is part of $\beta^{L}$ and one boundary
arc which is part of $\beta^{R}$. Each bead has two marked boundary points,
where these two arcs meet. Conditional on $\beta^{L}$ and $\beta^{R}$, let
$\eta_{-}$ be a curve from 0 to $\infty$ in $\overline{U}_{-}$ which consists
of the concatenation of conditionally independent chordal space-filling SLEκ
curves between the marked points of the beads of $U_{-}$. Similarly define
$\eta_{+}$ in $\overline{U}_{+}$. Then, let $\eta$ be the concatenation of the
time reversal of $\eta_{-}$ followed by $\eta_{+}$.

Figure 10:  Illustration of the construction of whole-plane space-filling SLE
from the chordal version used in the proof of Proposition 5.2, in the cases
$\kappa\geq 8$ and $\kappa\in(4,8)$.

The re-rooted curve $\eta^{0}$ is obtained by concatenating $\eta_{+}$
followed by the time reversal of $\eta_{-}$. Let
$\phi:\mathbbm{C}\rightarrow\mathbbm{C}$ be a fractional linear transformation
taking 0 to $\infty$ (i.e., $\phi(w)=a/w$ for some non-zero
$a\in\mathbbm{C}$). We will now argue that when $\kappa=8$, the curve
$\phi\circ\eta^{0}$ agrees in law with $\eta$ modulo time parametrization.

  * •

By the reversibility of whole-plane SLE2 [MS17, Theorem 1.20] followed by the
reversibility of chordal SLE${}_{2}(-1;-1)$ [MS16b, Theorem 1.1], the time
reversals of $\phi\circ\beta^{L}$ and $\phi\circ\beta^{R}$ have the same joint
law as $(\beta^{L},\beta^{R})$, viewed as curves modulo time parametrization.
In particular, $(\phi(U_{+}),\phi(U_{-}))\overset{d}{=}(U_{-},U_{+})$.

  * •

By the reversibility of chordal SLE8 [MS17, Theorem 1.19], the conditional law
given $(\beta^{L},\beta^{R})$ of the time reversals of $\phi\circ\eta_{-}$ and
$\phi\circ\eta_{+}$ is that of a pair of conditionally independent chordal
SLE8’s from 0 to $\infty$ in $\phi(U_{-})$ and $\phi(U_{+})$, respectively.

  * •

Consequently, concatenating $\phi\circ\eta_{+}$ followed by the time reversal
of $\phi\circ\eta_{-}$ gives us a new curve with the same law as $\eta$,
viewed modulo time parametrization.

When $\kappa\not=8$, whole-plane SLE${}_{16/\kappa}(16/\kappa-2)$ and chordal
SLE${}_{16/\kappa}(-8/\kappa;-8/\kappa)$ are still reversible (by the same
references), but chordal space-filling SLEκ is not [MS17, Theorem 1.19].
Hence, the exact same argument as above shows that $\phi\circ\eta^{0}$ does
not agree in law with $\eta$ modulo time parametrization when $\kappa\not=8$.
∎

###  5.2 Re-rooting invariance for SLE8-decorated $\gamma$-LQG

Using Proposition 5.2, we can obtain the following similar symmetry property
for SLE8-decorated LQG. The property is most easily stated in terms of curve-
decorated LQG surfaces, which we now define building on Definition 2.1

######  Definition 5.3.

Let $\gamma\in(0,2)$. A $\gamma$-LQG surface with a single marked point,
decorated by two curves is an equivalence class of 5-tuples
$(U,h,z,\eta_{1},\eta_{2})$, where

  * •

$U\subset\mathbbm{C}$ is open and $z\in U\cup\partial U$, with $\partial U$
viewed as a collection of prime ends;

  * •

$h$ is a generalized function on $U$ (which we will always take to be some
variant of the Gaussian free field);

  * •

$\eta_{1}:[a_{1},b_{1}]\rightarrow U\cup\partial U$ and
$\eta_{2}:[a_{2},b_{2}]\rightarrow U\cup\partial U$ are curves in
$U\cup\partial U$;

  * •

$(U,h,z,\eta_{1},\eta_{2})$ and
$(\widetilde{U},\widetilde{h},\widetilde{z},\widetilde{\eta}_{1},\widetilde{\eta}_{2})$
are declared to be equivalent if there is a conformal map
$\phi:\widetilde{U}\rightarrow U$ such that

$\widetilde{h}=h\circ\phi+Q\log|\phi^{\prime}|,\quad\phi(\widetilde{z})=z,\quad\phi\circ\widetilde{\eta}_{1}=\eta_{1}\quad\text{and}\quad\phi\circ\widetilde{\eta}_{2}=\eta_{2}$

where $Q$ is as in (2.5).

######  Proposition 5.4 (Re-rooting invariance for SLE8-decorated
$\gamma$-LQG).

Let $\gamma\in(0,2)$ and $\kappa_{1},\kappa_{2}>4$. Let
$(\mathbbm{C},h,\infty)$ be a singly marked unit-area quantum sphere. Also let
$\eta_{1},\eta_{2}$ be two independent whole-plane space-filling SLE curves
from $\infty$ to $\infty$, with parameters $\kappa_{1},\kappa_{2}$,
respectively, sampled independently from $h$ and then parametrized by
$\gamma$-LQG mass with respect to $h$. Fix $t\in[0,1]$. If
$\kappa_{1}=\kappa_{2}=8$ (and $\gamma\in(0,2)$ is arbitrary), then we have
the re-rooting invariance property

$\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\eta_{1}(t),\eta_{1}^{\eta_{1}(t)},\eta_{2}^{\eta_{1}(t)}}\right)\overset{d}{=}\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\infty,\eta_{1},\eta_{2}}\right)$
(5.2)

as curve-decorated quantum surfaces. This property is not true if either
$\kappa_{1}$ or $\kappa_{2}$ is not equal to 8.

Before giving the proof of Proposition 5.4, we explain why it implies Theorem
1.21.

###### Proof of Theorem 1.21 assuming Proposition 5.4.

For $t\in[0,1]$, define the permuton ${\boldsymbol{\pi}}_{t}$ as in the
theorem statement. Almost surely, $\eta_{1}(t)$ it not a multiple point for
$\eta_{1}$. Furthermore, by Lemma 2.10, a.s. $\psi(t)$, as defined in the
theorem statement, is the unique time at which $\eta_{2}$ hits $\eta_{1}(t)$.
Hence, by Definition 5.1 of the re-rooting operation,

$\eta_{1}(u)=\eta_{1}^{\eta_{1}(t)}\mathopen{}\mathclose{{}\left(u-t-\lfloor
u-t\rfloor}\right)\quad\text{and}\quad\eta_{2}(u)=\eta_{2}^{\eta_{1}(t)}\mathopen{}\mathclose{{}\left(u-\psi(t)-\lfloor
u-\psi(t)\rfloor}\right),\quad\forall u\in[0,1].$ (5.3)

By (5.3), Lemma 2.8, and the definition of ${\boldsymbol{\pi}}_{t}$, a.s. for
every rectangle $[a,b]\times[c,d]\subset[0,1]^{2}$,

${\boldsymbol{\pi}}_{t}\mathopen{}\mathclose{{}\left([a,b]\times[c,d]}\right)=\mu_{h}\mathopen{}\mathclose{{}\left(\eta_{1}^{\eta_{1}(t)}\mathopen{}\mathclose{{}\left([a,b]}\right)\cap\eta_{2}^{\eta_{1}(t)}\mathopen{}\mathclose{{}\left([c,d]}\right)}\right).$
(5.4)

By Proposition 5.4, we therefore have
${\boldsymbol{\pi}}_{t}\overset{d}{=}{\boldsymbol{\pi}}$. ∎

It remains to prove Proposition 5.4. We first prove the re-rooting invariance
property of Proposition 5.4 when we re-root at a uniformly random time, rather
than a deterministic time. This re-rooting invariance property is easier to
prove because if $T\in[0,1]$ is sampled uniformly from Lebesgue measure,
independently from everything else, then $\eta(T)$ is a sample from $\mu_{h}$,
so we can apply Lemma 2.3 and Proposition 5.2.

######  Lemma 5.5 (Re-rooting invariance for SLE8-decorated $\gamma$-LQG,
random time).

Let $(\mathbbm{C},h,\infty)$, $\eta_{1}$, and $\eta_{2}$ be as in Proposition
5.4. Let $T$ be sampled uniformly from Lebesgue measure on $[0,1]$,
independently from everything else. If $\kappa_{1}=\kappa_{2}=8$ (and
$\gamma\in(0,2)$ is arbitrary), then

$\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\eta_{1}(T),\eta_{1}^{\eta_{1}(T)},\eta_{2}^{\eta_{1}(T)}}\right)\overset{d}{=}\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\infty,\eta_{1},\eta_{2}}\right)$
(5.5)

as curve-decorated quantum surfaces. This property is not true if either
$\kappa_{1}$ or $\kappa_{2}$ is not equal to 8.

###### Proof.

Write $Z=\eta_{1}(T)$. Since $\eta_{1}$ is parametrized by $\gamma$-LQG mass
$\mu_{h}$ with respect to $h$, the point $Z$ is a uniformly sample from
$\mu_{h}$, independent from $\eta_{1}$ and $\eta_{2}$. Since the marked point
for a single marked quantum sphere is sampled uniformly from its LQG area
measure (Lemma 2.3), we have
$(\mathbbm{C},h,Z)\overset{d}{=}(\mathbbm{C},h,\infty)$ as quantum surfaces.

Since $(h,Z)$ is independent from $\eta_{1}$ and $\eta_{2}$, viewed modulo
time parametrization, Proposition 5.2 shows that when
$\kappa_{1}=\kappa_{2}=8$, the conditional law of
$(\eta_{1}^{Z},\eta_{2}^{Z})$ given $(h,Z)$ is that of a pair of conditionally
independent whole-plane SLE8 curves from $Z$ to $Z$. Moreover, these curves
are parametrized by $\mu_{h}$-mass. Therefore, (5.5) holds when
$\kappa_{1}=\kappa_{2}=8$.

If, say, $\kappa_{1}\not=8$, then Proposition 5.2 shows that the conditional
law of $\eta_{1}^{Z}$ given $(h,Z)$ is _not_ that of a whole-plane space-
filling SLE${}_{\kappa_{1}}$ from $Z$ to $Z$. Therefore, (5.5) does not hold
if $\kappa_{1}\not=8$. The same argument also works if $\kappa_{2}\not=8$. ∎

###### Proof of Proposition 5.4.

The re-rooting invariance property (5.2) at a fixed deterministic time
immediately implies the re-rooting invariance property (5.5) at a time $T$
sampled uniformly from $[0,1]$. Hence, Lemma 5.5 implies that (5.2) does not
hold if either $\kappa_{1}$ or $\kappa_{2}$ is not equal to 8. We need to
prove that (5.2) holds when $\kappa_{1}=\kappa_{2}=8$.

Consider the space of 5-tuples $(\mathbbm{C},\mathfrak{h},z,P_{1},P_{2})$
where $\mathfrak{h}$ is a generalized function,
$z\in\mathbbm{C}\cup\\{\infty\\}$, and
$P_{1},P_{2}:[0,1]\rightarrow\mathbbm{C}\cup\\{\infty\\}$ are curves such that
$P_{1}(0)=P_{1}(1)$, $P_{1}([0,1])=\mathbbm{C}\cup\\{\infty\\}$, and the same
is true for $P_{2}$. For $t\in[0,1]$, we define the re-rooting operator on
this space of 5-tuples by

$\mathcal{R}_{t}\mathopen{}\mathclose{{}\left(\mathbbm{C},\mathfrak{h},z,P_{1},P_{2}}\right)=\mathopen{}\mathclose{{}\left(\mathbbm{C},\mathfrak{h},P_{1}(t),P_{1}^{P_{1}(t)},P_{2}^{P_{1}(t)}}\right),$
(5.6)

here using the notation of Definition 5.1. From (5.1), one can check that if
$s,t\in[0,1]$ and neither $P_{1}(t)$ nor $P_{1}([s+t])$ is hit more than once
by either $P_{1}$ or $P_{2}$, then

$\mathcal{R}_{s}\circ\mathcal{R}_{t}=\mathcal{R}_{[s+t]},\quad\text{where}\quad[s+t]:=s+t-\lfloor
s+t\rfloor.$ (5.7)

Now let $t\in[0,1]$ be fixed and let $T$ be sampled uniformly from Lebesgue
measure on $[0,1]$, independently from everything else. Then $[t+T]$ is also
sampled uniformly from Lebesgue measure on $[0,1]$ independently from
everything else. Therefore, if $\kappa_{1}=\kappa_{2}=8$ then Lemma 5.5 gives

$\mathcal{R}_{T}\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\infty,\eta_{1},\eta_{2}}\right)\overset{d}{=}\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\infty,\eta_{1},\eta_{2}}\right)\overset{d}{=}\mathcal{R}_{[t+T]}\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\infty,\eta_{1},\eta_{2}}\right),$
(5.8)

as curve-decorated quantum surfaces. Since $\eta_{1}(T)$ and $\eta_{1}([t+T])$
are sampled uniformly from $\mu_{h}$, independently from $\eta_{1}$ and
$\eta_{2}$, a.s. each of $\eta_{1}(T)$ and $\eta_{1}([t+T])$ is hit only once
by each of $\eta_{1}$ and $\eta_{2}$. So, we can apply (5.7) followed by the
second equality in (5.8) to obtain

$\mathcal{R}_{t}\circ\mathcal{R}_{T}\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\infty,\eta_{1},\eta_{2}}\right)\overset{d}{=}\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\infty,\eta_{1},\eta_{2}}\right).$
(5.9)

By the first equality in (5.8), the equality in law (5.9) also holds with