$\mathcal{R}_{t}\circ\mathcal{R}_{T}(\mathbbm{C},h,\infty,\eta_{1},\eta_{2})$
replaced by
$\mathcal{R}_{t}\mathopen{}\mathclose{{}\left(\mathbbm{C},h,\infty,\eta_{1},\eta_{2}}\right)$
on the left. That is, (5.2) holds. ∎

##  6 Physics background for the meander conjecture

In this section we review the developments in statistical physics that lead to
Conjecture 1.3, namely, the scaling limit of meanders is described by LQG with
matter central charge ${\mathbf{c}_{\mathrm{M}}}=-4$, decorated by two
independent SLE8 curves. Section 6.1 gives a quick overview of the ideas that
lead to Conjecture 1.3, then the consecutive sections review in more detail
the reasoning. At the end of the section (Section 6.6), we will also give a
heuristic justification of Conjecture 1.8 using Conjecture 1.3.

###  6.1 Meanders, Hamiltonian cycles, and spanning trees

The scaling limits of a large number of 2D statistical physics models are
expected to be described by conformal field theories (CFTs); see [ISZ98] for a
review. A key characteristic of a CFT is its central charge
${\mathbf{c}}\in\mathbbm{R}$, which encodes how its partition function
responds to deformations of the background metric. Although the full field-
theoretic description of the scaling limit has only been achieved for a
restricted class of models such as the critical Ising model [BPZ84], for a
wide range of models, the central charge can be derived from the asymptotic
behavior of the partition function of the model on periodic lattices. For
example, by Kirchhoff’s matrix tree theorem, the scaling limit of the uniform
spanning tree on a large class of 2D lattices should be a CFT with central
charge ${\mathbf{c}}=-2$.

The Peano curve of a spanning tree on a 2D lattice is equivalent to the
Hamiltonian cycle on a related directed lattice [Kas63], which means that the
uniform directed Hamiltonian cycle on a large class of directed lattices
should also have central charge ${\mathbf{c}}=-2$. However, for the uniform
undirected Hamiltonian cycle on the honeycomb lattice, the central charge
turns out to be ${\mathbf{c}}=-1$ instead of ${\mathbf{c}}=-2$ [BN94, BSY94].
The honeycomb lattice is an example of a cubic planar map whose dual graph is
a triangulation. As explained in [GKN99], the central charge discrepancy
between directed and undirected Hamiltonian cycles is due to a special
property of the honeycomb lattice, that is, its dual graph is an Eulerian
triangulation (i.e. each vertex has an even degree). Since this property does
not hold for a typical cubic planar map (whose dual is a triangulation), the
uniform random triangulation decorated by a Hamiltonian cycle on its dual
should be in the same universality class as a uniform spanning tree decorated
map. The scaling limit of this latter decorated map is expected to be
$\gamma$-LQG with matter central charge ${\mathbf{c}_{\mathrm{M}}}=-2$
(equivalently, $\gamma\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\sqrt{2}$; recall the discussion in Remark 2.6), decorated by an
SLE8 curve; see, e.g., [She16b, Appendix A].

A meander can be viewed as a pair of Hamiltonian cycles on a planar map whose
dual is an Eulerian quadrangulation, with the additional constraint that at
each vertex the two cycles cross each other. Following the line of reasoning
as in [GKN99] with a crucial new insight, Di Francesco, Golinelli, and Guitter
[DFGG00] argued that the uniformly sampled meander should be in the same
universality class as random planar maps decorated by a _pair_ of independent
spanning trees. Since the central charge is believed to be additive for
independent models, such planar maps are expected to converge to LQG with
matter central charge ${\mathbf{c}_{\mathrm{M}}}=(-2)+(-2)=-4$ (equivalently,
$\gamma\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\sqrt{\frac{1}{3}\mathopen{}\mathclose{{}\left(17-\sqrt{145}}\right)}$),
decorated by a pair of independent SLE8 curves; as stated in Conjecture 1.3.

In the rest of this section, we review in more detail the reasoning of
[DFGG00]. The key is a subtle relation between the fully packed and the dense
O$(n)$ loop model on different lattices. See Figure 11 for a summary of the
models discussed in the following sections.

Figure 11: A table summarizing the discussions on the O$(n)$ and related
models in Section 6. For each box (i.e. for each model), above the dotted line
there is the expression for the central charge of the associated CFT; below
the dotted line there is the corresponding scaling limit. The two bottom boxes
summarize the conjectures explained in Sections 6.4 and 6.5 for the
corresponding planar maps models. We recall that the relation between $\gamma$
and ${\mathbf{c}}_{M}$ is given in (2.12). A legend for the use of the colors
is given on the top right: recall that the definitions of _matched_ and
_mismatched_ models are given in Remark 1.4.

###  6.2 Positive temperature O($n$) loop model and its scaling limit

For a fixed $n>0$, the O($n$) loop model on a graph is the statistical physics
model where a configuration $\mathcal{L}$ is a collection of disjoint loops
(i.e. simple cycles). Given the temperature $T>0$, the weight of each
configuration is $T^{-\mathcal{V}(\mathcal{L})}n^{\mathcal{N}(\mathcal{L})}$,
where $\mathcal{V}(\mathcal{L})$ is the total number of vertices occupied by
all loops in $\mathcal{L}$ and $\mathcal{N}(\mathcal{L})$ is the total number
of loops in $\mathcal{L}$. When $n\in(0,2]$, for the O($n$) loop model on a
planar lattice, there exists a critical temperature $T_{c}>0$ such that the
following is true. For $T=T_{c}$, the scaling limit of the O$(n)$ loop model
is conjecturally described by a CFT of central charge

${\mathbf{c}}_{\textrm{dilute}}(n):=1-6\mathopen{}\mathclose{{}\left(\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2}}\right)^{2},\quad\text{where
$\kappa\in(8/3,4]$ solves
$n=2\cos\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(1-\frac{4}{\kappa}}\right)\pi}\right)$}.$
(6.1)

For $0<T<T_{c}$, the scaling limit is instead conjecturally described by a CFT
with central charge

${\mathbf{c}}_{\textrm{dense}}(n)=1-6\mathopen{}\mathclose{{}\left(\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2}}\right)^{2},\quad\text{where
$\kappa\in[4,8)$ solves
$n=2\cos\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(1-\frac{4}{\kappa}}\right)\pi}\right)$}.$
(6.2)

These two regimes are called the dilute and dense phases, respectively. See
the lecture notes [Jac09] for more background for the relation between CFT and
the O$(n)$ loop model.

After the discovery of SLE [Sch00], it was further conjectured that the
collection of loops in the dilute and the dense O($n$) loop models converge to
the conformal loop ensemble whose parameter $\kappa\in(\frac{8}{3},8)$ is
related to $n$ as above (a conformal loop ensemble (CLE) is a conformally
invariant collection of SLEκ-type loops [She09]). For $n=0$, the O$(n)$ loop
model is supported on configurations with $\mathcal{N}(\mathcal{L})=1$, whose
weight is $T^{-\mathcal{V}(\mathcal{L})}$. For $T=T_{c}$, the model becomes
the (critical) self-avoiding polygon and the conjectural scaling limit of the
single loop is SLE8/3. For $T\in(0,T_{c})$ the conjectural limit is SLE8,
which coincides with the scaling limit of the uniform spanning tree [LSW04].
The CFT central charges in the two cases are
${\mathbf{c}}_{\textrm{dilute}}(0)=\lim_{n\to 0}c_{\textrm{dilute}}(n)=0$ and
${\mathbf{c}}_{\textrm{dense}}(0)=\lim_{n\to 0}c_{\textrm{dense}}(n)=-2$,
respectively. All the aforementioned scaling limit conjectures are believed to
be independent of the underlying lattice. See [PS17] for more background on
the O$(n)$ loop model. See [Smi07, Section 2.2], [She09, Section 2.3] and
references therein for the connection between $O(n)$ loop models and CLE.

###  6.3 Fully packed O($n$) loop model and its scaling limit

As $T\to 0$, the O($n$) loop model is supported on configurations that
maximize $\mathcal{V}(\mathcal{L})$. This is the so-called fully packed O($n$)
loop model. In contrast to the dilute and dense phases, the scaling limit of
the fully packed O($n$) loop model highly depends on the underlying lattice.
In particular, on the honeycomb lattice, Blote and Nienhuis [BN94] argued that
on the one hand, the scaling limit of the loops in the fully packed phase is
in the same universality class as those in the dense phase, i.e., it should be
described by the same CLE. On the other hand, the CFT descibing the scaling
limit has central charge

${\mathbf{c}}_{\textrm{fp}}(n)={\mathbf{c}}_{\textrm{dense}}(n)+1,$ (6.3)

where ${\mathbf{c}}_{\textrm{dense}}$ is as in (6.2).

The physics argument for (6.3) is the following. On the honeycomb lattice, a
loop configuration $\mathcal{L}$ maximizes $\mathcal{V}(\mathcal{L})$ when it
covers all vertices. Therefore, the fully packed O$(1)$ loop model is the
uniform measure on such configurations. By [BN94], a fully packed
configuration $\mathcal{L}$ on the honeycomb lattice can be mapped to an
integer-valued function $h(\mathcal{L})$ on the dual triangular lattice, which
we call the height function. Under this mapping, the fully packed O$(1)$ loop
model on the honeycomb lattice is equivalent to a solid-on-solid (SOS) model
which produces the same height function. (Explaining the height function
construction would be too much of a digression; see [GKN99, Figure 1] and
[NHB84, Figure 2-4] for demonstrations.) On the other hand, thanks to a
connection to the zero-temperature triangular anti-ferromagnetic Ising model,
it was already known from [BH82, BN93] that the SOS model has central charge
${\mathbf{c}}=1$, and the height function $h(\mathcal{L})$ is believed to
converge to the Gaussian free field. This means that
${\mathbf{c}}_{\textrm{fp}}(n=1)=1$. Since
${\mathbf{c}}_{\textrm{dense}}(n=1)=0$ by (6.2), we get (6.3) when $n=1$. For
$n\neq 1$, although $h(\mathcal{L})$ is determined by $\mathcal{L}$, it is
conjectured in [BN94] that $\mathcal{L}$ and $h(\mathcal{L})$ are
asymptotically decoupled and their joint scaling limit is equivalent to that
of the the superposition of the dense O$(n)$ loop model and the
${\mathbf{c}}=1$ SOS model. Since the central charge is additive for
independent models, (6.3) holds for all $n\in[0,2]$. This relation is
supported by some numerical evidence in [BN94]. It was put on a solid ground
at the physics level of rigor via the exact Bethe ansatz techniques in
[BSY94].

For $n=0$, the fully packed loop model on the honeycomb lattice is supported
on configurations where a single loop visits each vertex, namely, it produces
a uniformly sampled Hamiltonian cycle on the honeycomb lattice. According to
(6.3), the central charge of the corresponding CFT is
${\mathbf{c}}_{\textrm{fp}}(0)=\lim_{n\to
0}{\mathbf{c}}_{\textrm{dense}}(n)+1=-1$. Moreover, the scaling limit of the
Hamiltonian cycle should be an SLE8 curve (which is the limit of CLEκ as
$\kappa\nearrow 8$).

###  6.4 Fully packed O($n$) loop model on random planar maps

It was observed by Guitter, Kristjansen, and Nielsen [GKN99] that the height
function $h(\mathcal{L})$ for a fully packed loop configuration $\mathcal{L}$
does not necessarily make sense if the honeycomb lattice is replaced by an
arbitrary planar map whose dual map is a triangulation. It makes sense if and
only if the triangulation is Eulerian.999We recall for later convenience the
following equivalent characterizations of a Eulerian triangulation: (1) a
planar map such that all its faces are triangles and the degree of each vertex
is even; (2) a planar map such that all its faces are triangles, colored in
black and white, so that no two adjacent faces have the same color; (3) a
planar map such that its dual map is a planar bicubic map, i.e. all its
vertices have degree three and these vertices are colored in black and white
so that any two adjacent vertices have different colors. Recalling the
conjectural connection between LQG and random planar maps (Remark 2.6), this
leads to the conjecture that the scaling limit of a uniform random Eulerian
triangulation decorated by a Hamiltonian cycle (i.e. by a fully packed
O($n=0$) loop model) on its dual map is described by $\gamma$-LQG with matter
central charge

${\mathbf{c}_{\mathrm{M}}}={\mathbf{c}}_{\textrm{dense}}(n=0)+1=-1$

(equivalently, $\gamma\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\sqrt{\frac{2}{3}(7-\sqrt{13})}=\frac{\sqrt{13}-1}{\sqrt{3}}$),
decorated by an SLE8 curve. Via the KPZ relation [KPZ88], it was conjectured
and numerically tested in [GKN99] that the number of (rooted) Hamiltonian-
cycle decorated Eulerian triangulations of size $n$ behaves asymptotically
like $A^{n}n^{-\alpha}$ with
$\alpha\stackrel{{\scriptstyle\eqref{eq:exp}}}{{=}}1+4/\gamma^{2}=(13+\sqrt{13})/6$.

If we consider an ordinary (not necessarily Eulerian) triangulation decorated
by a Hamiltonian cycle (i.e. by a fully packed O($n=0$) loop model) on its
dual map, Guitter, Kristjansen, and Nielsen [GKN99] argued that since the $+1$
increment in (6.3) disappears, the scaling limit should be $\gamma$-LQG with
matter central charge
${\mathbf{c}_{\mathrm{M}}}={\mathbf{c}}_{\textrm{dense}}(n=0)=-2$
(equivalently, $\gamma\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\sqrt{2}$). In addition to the matrix model justification in
[EGK98], we sketch a justification of this assertion using mating of trees
[DMS21]. The dual map of a triangulation with a Hamiltonian cycle can be
visualized as a picture similar to Figure 3, where the Hamiltonian cycle is a
straight line and other edges on the dual map are on the two sides of the
straight line (in this setting, the other edges do not necessarily form a
loop). This produces a pair of non-crossing matchings that can be viewed as
two planar trees. They converge to a pair of independent continuum random
trees. By [DMS21, Theorem 1.9], the mating of these trees gives LQG with
${\mathbf{c}_{\mathrm{M}}}=-2$ (equivalently,
$\gamma\stackrel{{\scriptstyle\eqref{eqn:lqg-ccM}}}{{=}}\sqrt{2}$) decorated
by an SLE8 curve which describes the scaling limit of the Hamiltonian cycle
(i.e., the interface between the two trees).

######  Remark 6.1.

Although it is not needed for the justification of Conjecture 1.3, we note
that, as explained in Section 3.3 of the recent paper [DDGG22], the
conjectures in this section can be extended to a triangulation decorated by
the fully packed O($n$) loop model on its dual map for $n>0$. In particular,
for an Eulerian triangulation decorated by a fully packed O($n$) loop model on
its dual map for $n\in(0,1]$,101010As explained in [DDGG22, Section 3.3] in
the case of a uniform random Eulerian triangulation one can consider the fully
packed O($n$) loop model only in the range $n\in[0,1]$. Indeed, the case
$n\in(1,2]$ should degenerate in the scaling limit to a tree-like structure.
the scaling limit should be described by $\gamma(n)$-LQG with matter central
charge

${\mathbf{c}_{\mathrm{M}}}(n)={\mathbf{c}}_{\textrm{dense}}(n)+1\stackrel{{\scriptstyle\eqref{eq:cdense}}}{{=}}2-6\mathopen{}\mathclose{{}\left(\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2}}\right)^{2},\quad\text{where
$\kappa\in[6,8)$ solves
}n=2\cos\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(1-\frac{4}{\kappa}}\right)\pi}\right)$

(equivalently, $\gamma(n)\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\frac{1}{\sqrt{12}}\mathopen{}\mathclose{{}\left(\sqrt{3\mathopen{}\mathclose{{}\left(\kappa+\frac{16}{\kappa}}\right)+22}-\sqrt{3\mathopen{}\mathclose{{}\left(\kappa+\frac{16}{\kappa}}\right)-26}}\right)\in\mathopen{}\mathclose{{}\left(\frac{\sqrt{13}-1}{\sqrt{3}},2}\right]$),
decorated by a CLEκ(n). (Recall that the limit of CLEκ as $\kappa\nearrow 8$
is SLE8, making this conjecture consistent with the $n=0$ case described
above.) For an ordinary (not necessarily Eulerian) triangulation decorated by
a fully packed O($n$) loop model on its dual map for $n\in(0,2]$, the scaling
limit should be described by $\gamma(n)$-LQG with matter central charge

${\mathbf{c}_{\mathrm{M}}}(n)={\mathbf{c}}_{\textrm{dense}}(n)\stackrel{{\scriptstyle\eqref{eq:cdense}}}{{=}}1-6\mathopen{}\mathclose{{}\left(\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2}}\right)^{2},\quad\text{where
$\kappa\in[4,8)$ solves
}n=2\cos\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(1-\frac{4}{\kappa}}\right)\pi}\right)$

(equivalently, $\gamma(n)\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\frac{4}{\sqrt{k}}\in(\sqrt{2},2]$), decorated by a CLEκ(n). Note
that $\gamma$ and $\kappa$ are mismatched (in the sense of Remark 1.4) for
random Eulerian triangulations decorated by a fully packed O($n$) loop model
on their dual map for $n\in[0,1]$, and are matched for ordinary triangulations
decorated by a fully packed O($n$) loop model on their dual map for
$n\in[0,2]$.

###  6.5 Meanders and the fully packed O$(n\times m)$ loop model

Di Francesco, Golinelli, and Guitter [DFGG00] arrived at their meander
conjecture following a similar reasoning as in [GKN99]. They first considered
the (double) fully packed O$(n\times m)$ loop model on the square lattice
$\mathbbm{Z}^{2}$ with $n,m\in[0,2]^{2}$. In this model, each configuration is
a pair of loop collections $(\mathcal{L}_{1},\mathcal{L}_{2})$ such that every
edge is visited by either a loop in $\mathcal{L}_{1}$ or a loop in
$\mathcal{L}_{2}$ but not both, while every vertex is visited by both types of
loops. For $n,m\in(0,2]^{2}$, the weight of each configuration is
$n^{\mathcal{N}(\mathcal{L}_{1})}m^{\mathcal{N}(\mathcal{L}_{2})}$, where, as
above, $\mathcal{N}(\mathcal{L}_{i})$ is the number of loops in
$\mathcal{L}_{i}$ for $i=1,2$. For $n=m=0$, the model is supported on
configurations with
$\mathcal{N}(\mathcal{L}_{1})=\mathcal{N}(\mathcal{L}_{2})=1$ and assigns
equal weight to them.

Again using a mapping to a height function model, Jacobsen and Kondev [JK98,
JK99] made a similar conjecture as (6.3) on the central charge
${\mathbf{c}}_{\textrm{dfp}}(n,m)$ of the scaling limit of the fully packed
O$(n\times m)$ loop model on $\mathbbm{Z}^{2}$. In order to define such a
mapping to a height function model, Jacobsen and Kondev first considered the
uniform measure on configurations
$(\vec{\mathcal{L}_{1}},\vec{\mathcal{L}_{2}})$ on $\mathbbm{Z}^{2}$, where
$(\vec{\mathcal{L}_{1}},\vec{\mathcal{L}_{2}})$ are two collections of
oriented loops satisfying the same conditions above. As explained in [DFGG00,
Section 3], Jacobsen and Kondev [JK98, JK99] mapped
$(\vec{\mathcal{L}_{1}},\vec{\mathcal{L}_{2}})$ on $\mathbb{Z}^{2}$ to a 3D
height function
$(h_{1}(\vec{\mathcal{L}_{1}},\vec{\mathcal{L}_{2}}),h_{2}(\vec{\mathcal{L}_{1}},\vec{\mathcal{L}_{2}}),h_{3}(\vec{\mathcal{L}_{1}},\vec{\mathcal{L}_{2}}))$,
which they conjectured to converge to three independent Gaussian free fields,
hence a conformal theory with central charge $c=3$. Introducing some specific
weights for the oriented configurations
$(\vec{\mathcal{L}_{1}},\vec{\mathcal{L}_{2}})$, Jacobsen and Kondev also
determined a general expression for the central charge of the scaling limit of
this new weighted oriented model (see [JK98, Eq. (6.4)]). Using a simple
relation (given in [JK98, Eq. (4.1)]) between the weights of the oriented
configurations $(\vec{\mathcal{L}_{1}},\vec{\mathcal{L}_{2}})$ and the weights
$(n,m)$ of the fully packed O$(n\times m)$ loop model, i.e. of the unoriented
configurations $(\mathcal{L}_{1},\mathcal{L}_{2})$, they finally conjectured
that the central charge of the scaling limit of the fully packed O$(n\times
m)$ loop model on $\mathbbm{Z}^{2}$ is given by:

${\mathbf{c}}_{\textrm{dfp}}(n,m)={\mathbf{c}}_{\textrm{dense}}(n)+{\mathbf{c}}_{\textrm{dense}}(m)+1.$
(6.4)

They also conjectured that $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$
asymptotically decouple, i.e., their scaling limits are independent. The
relation in (6.4) has also been justified at a physics level of rigor via the
exact Bethe ansatz techniques [DCN04, JZJ04]. For $n=m=0$, the condition
$\mathcal{N}(\mathcal{L}_{1})=\mathcal{N}(\mathcal{L}_{2})=1$ means that we
have two Hamiltonian cycles on the square lattice whose edge sets are
disjoint. According to (6.4), the central charge of the limiting CFT is
${\mathbf{c}}_{\textrm{dfp}}(0,0)=2\,{\mathbf{c}}_{\textrm{dense}}(0)+1=-3$.
Moreover, the two Hamiltonian cycles should converge to two independent SLE8
curves (following the same reasoning in the discussion after (6.2)).

Recall from Section 1.3 that a meander can be viewed as a triplet
$(M,P^{1},P^{2})$, where $P^{1},P^{2}$ are two Hamiltonian cycles with
disjoint edge sets on a 4-regular planar map $M$, i.e. a planar map with all
vertices of degree four. Moreover, at each vertex the two cycles $P^{1}$ and
$P^{2}$ cross each other. This implies that the edges on the boundary of each
face of the dual map of $M$ alternate between dual edges which cross edges of
$P^{1}$ and dual edges which cross edges of $P^{2}$. Hence, each face of $M$
has even degree, and so the dual of $M$ is an Eulerian quadrangulation.

As explained in [DFGG00, Section 3], the height function mapping of Jacobsen
and Kondev described in the previous section makes sense on a general planar
map $G$ (instead of $\mathbb{Z}^{2}$) if and only if the dual map of $G$ is an
Eulerian quadrangulation. Since the dual of $M$ is an Eulerian
quadrangulation, one can associate to $(M,P^{1},P^{2})$ a height function
$(h_{1},h_{2},h_{3})$ as above. However, the additional constraint that the
two cycles must cross at each vertex yields that $(h_{1},h_{2},h_{3})$ must
lie on a 2D subspace. This effect exactly cancels the $+1$ in (6.4). (See
[DFGG00, Figure 2] and the discussion below that figure.) Therefore, the
scaling limit should be $\gamma$-LQG with
${\mathbf{c}_{\mathrm{M}}}=2\,{\mathbf{c}}_{\operatorname{dense}}(0)=-4$
(equivalently, $\gamma\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\sqrt{\frac{1}{3}(17-\sqrt{145})}$), decorated by two independent
SLE8 curves as in Conjecture 1.3. This leads to several enumeration
conjectures in [DFGG00] on meanders which are supported by numerical evidence
in the follow-up works [DFGJ00, JG00]. In particular, they conjectured that
the number of meanders grows as $A^{n}n^{-\alpha}$, where $A\approx 12.26$ and
$\alpha\stackrel{{\scriptstyle\eqref{eq:exp}}}{{=}}1+4/\gamma^{2}=(29+\sqrt{145})/12$.

Although not mentioned explicitly in [DFGG00], their argument naturally leads
to the following scaling limit conjectures in addition to Conjecture 1.3.
Consider a uniform 4-regular planar map decorated by a pair of edge-disjoint
Hamiltonian cycles, without the constraint that the cycles cross at each
vertex. Such configurations are called _tangent meanders_ in [DFGJ00]. The
scaling limit should be described by $\gamma$-LQG with
${\mathbf{c}_{\mathrm{M}}}=-4$ (equivalently,
$\gamma\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\sqrt{\frac{1}{3}(17-\sqrt{145})}$). If we further condition on the
event that the dual quandrangulation is Eulerian, then we should instead get
$\gamma$-LQG with ${\mathbf{c}_{\mathrm{M}}}=-3$ (equivalently,
$\gamma\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}\sqrt{\frac{4}{3}(4-\sqrt{7})}$). Finally, consider a uniformly
sampled pair of Hamiltonian cycles on $\mathbb{Z}^{2}$ with the condition that
they form a meander. Then the central charge for the corresponding CFT should
be ${\mathbf{c}}=-4$. In all cases, the two Hamiltonian cycles should converge
to two independent SLE8 curves.

As in Remark 6.1, the same reasoning as in the conjecture for meanders can be
extended to get a scaling limit conjecture for a 4-regular planar map
decorated by the fully packed O$(n\times m)$ loop model for
$(n,m)\in[0,2]^{2}$, subject to the constraint that the loops cross at each
vertex. Recall that
${\mathbf{c}}_{\textrm{dense}}(n)\stackrel{{\scriptstyle\eqref{eq:cdense}}}{{=}}1-6\mathopen{}\mathclose{{}\left(\frac{2}{\sqrt{\kappa(n)}}-\frac{\sqrt{\kappa(n)}}{2}}\right)^{2}$,
where $\kappa(n)\in[4,8]$ solves
$n=2\cos\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(1-\frac{4}{\kappa(n)}}\right)\pi}\right)$.
The following conjecture follows from exactly the same reasoning as the
meander conjecture above.

######  Conjecture 6.2.

For $k\in\mathbb{N}$, let $\mathcal{C}_{k}$ be the set of triples
$(M,\mathcal{L}_{1},\mathcal{L}_{2})$, where $M$ is a 4-regular planar map
with $2k$ vertices; and $(\mathcal{L}_{1},\mathcal{L}_{2})$ is a pair of
collections of loops on $M$ with the following properties:

  * •

every edge of $M$ is visited by exactly one loop in
$\mathcal{L}_{1}\cup\mathcal{L}_{2}$;

  * •

every vertex of $M$ is visited by exactly one loop in $\mathcal{L}_{1}$ and
exactly one loop in $\mathcal{L}_{2}$;

  * •

every pair of loops
$(\ell_{1},\ell_{2})\in\mathcal{L}_{1}\times\mathcal{L}_{2}$ cross each other
at every shared vertex (i.e., the edges of $\ell_{1}$ and $\ell_{2}$ in cyclic
order around the vertex alternate).

Fix $(n,m)\in[0,2]^{2}$ such that111111If
${\mathbf{c}}_{\textrm{dense}}(n)+{\mathbf{c}}_{\textrm{dense}}(m)>1$, the
planar map should degenerate in the scaling limit to a tree-like structure.
${\mathbf{c}}_{\textrm{dense}}(n)+{\mathbf{c}}_{\textrm{dense}}(m)\leq 1$. Let
$(M,\mathcal{L}_{1},\mathcal{L}_{2})$ be sampled from the uniform measure on
$\mathcal{C}_{k}$ weighted by
$n^{\mathcal{N}(\mathcal{L}_{1})}m^{\mathcal{N}(\mathcal{L}_{2})}$. Then
$(M,\mathcal{L}_{1},\mathcal{L}_{2})$ converges under an appropriate scaling
limit to a Liouville quantum gravity sphere with matter central charge

${\mathbf{c}_{\mathrm{M}}}(n,m)={\mathbf{c}}_{\mathrm{dense}}(n)+{\mathbf{c}}_{\mathrm{dense}}(m),$

equivalently, with coupling constant $\gamma(n,m)$ satisfying
${\mathbf{c}_{\mathrm{M}}}(n,m)\stackrel{{\scriptstyle\eqref{eqn:lqg-
ccM}}}{{=}}25-6\,\mathopen{}\mathclose{{}\left(2/\gamma(n,m)+\gamma(n,m)/2}\right)^{2}$,
decorated by a whole-plane CLEκ(n) and an independent whole-plane CLEκ(m). If
$\kappa(n)=8$ or $\kappa(m)=8$ then CLE8 is interpreted as SLE8.

###  6.6 Convergence of the cyclic meandric permutation to the identity

Recall Conjecture 1.8, which asserts that the cyclic meandric permutation
$\tau_{\ell_{n}}$ associated with a uniform meander $\ell_{n}$ of size $n$
converges to the identity permuton. In this subsection we will give a
heuristic justification of Conjecture 1.8 using Conjecture 1.3. To prove
Conjecture 1.8, one would need to show that with high probability when $n$ is
large, $|\tau_{\ell_{n}}(j)-j|$ is of order $o(n)$ for all but all but a
negligible fraction of the values of $j\in[1,2n]\cap\mathbbm{Z}$.

To see why this should be true, consider the setting of Section 1.2 with
$\eta_{1},\eta_{2}$ independent SLE8 curves and
$\gamma=\sqrt{\frac{1}{3}(17-\sqrt{145})}$, so that ${\boldsymbol{\pi}}$ is
the meandric permuton. Let $\psi,\phi:[0,1]\rightarrow[0,1]$ be two Lebesgue
measurable function chosen so that $\eta_{1}(t)=\eta_{2}(\psi(t))$ and
$\eta_{1}(\phi(t))=\eta_{2}(t)$. We claim that a.s. for Lebesgue-a.e.
$t\in[0,1]$,

$\lim_{\varepsilon\rightarrow
0}\sup_{s\in[-\varepsilon,\varepsilon]}|\psi(\phi(t)+s)-t|=0.$ (6.5)

If we think of $\eta_{1}$ and $\eta_{2}$ as the continuum limits of the loop
and the line, respectively, for the meander $\ell_{n}$, then we can think of
$\psi$ and $\phi$ as the continuum analogs of $\sigma_{\ell_{n}}$ and
$\sigma_{\ell_{n}}^{-1}$, respectively. Hence, if we assume Conjecture 1.3,
then (6.5) should imply that for a typical $t\in[0,1]$,

$\lim_{\varepsilon\rightarrow
0}\lim_{n\rightarrow\infty}\sup_{k\in[-2\varepsilon n,2\varepsilon
n]\cap\mathbbm{Z}}\frac{1}{2n}\mathopen{}\mathclose{{}\left|\sigma_{\ell_{n}}(\sigma_{\ell_{n}}^{-1}(\lfloor
2tn\rfloor)+k)-2tn}\right|=0.$ (6.6)

By the formula
$\tau_{\ell_{n}}(j)=\sigma_{\ell_{n}}(\sigma_{\ell_{n}}^{-1}(j)+1)$ from
(1.3), the relation (6.6) for Lebesgue-a.e. $t\in[0,1]$ implies the desired
convergence of $\tau_{\ell_{n}}$ to the identity permuton.

We now explain why (6.5) is true. For $\mu_{h}$-a.e. $z\in\mathbbm{C}$, each
of $\eta_{1}$ and $\eta_{2}$ hits the point $z$ exactly once (c.f. the proof
of Lemma 2.8). Equivalently, for Lebesgue-a.e. $t\in[0,1]$, the each of
$\eta_{1}$ and $\eta_{2}$ hits $\eta_{2}(t)$ exactly once. For such a time
$t$, the point $\eta_{2}(t)$ is not on the boundary of
$\eta_{2}([t-\delta,t+\delta])$ for any $\delta>0$ (otherwise, $\eta_{2}$
would have to hit $t$ at a time not in $[t-\delta,t+\delta]$). Hence,
$\eta_{2}([t-\delta,t+\delta])$ contains a neighborhood of $\eta_{2}(t)$.
Since $\eta_{1}$ is a continuous curve and $\eta_{1}$ hits $\eta_{2}(t)$
exactly once (namely, at the time $\phi(t)$), there exists $\varepsilon>0$
such that

$\eta_{1}([\phi(t)-\varepsilon,\phi(t)+\varepsilon])\subset\eta_{2}([t-\delta,t+\delta])\quad\text{which
implies}\quad\sup_{s\in[-\varepsilon,\varepsilon]}|\psi(\phi(t)+s)-t|\leq\delta.$

Since $\delta>0$ is arbitrary, this gives (6.5).

##  7 Open problems

It is difficult to imagine how one would make a direct connection between a
subsequential limit of meandric permutons and SLE-decorated LQG. Consequently,
in order to prove Conjecture 1.7 one would likely first need to solve the
following problem.

######  Problem 7.1.

Is there a description of the meandric permuton which does not use SLE or LQG?

Possible solutions to Problem 7.1 might include an axiomatic characterization
of the meandric permuton or a direct construction of this permuton in terms of
Brownian motion (similar to the construction of the skew Brownian permutons in
[Bor21c]). We emphasize that Conjecture 1.7 is likely to be difficult to prove
even if one finds a nice solution to Problem 7.1, since meanders are
notoriously difficult to analyze.

It seems quite difficult to get a lower bound for the length of the longest
increasing subsequence for classes of random permutations which are known or
expected to converge to the random permutons considered in this paper.
Currently, the only available lower bounds are the essentially trivial ones,
namely
$\operatorname{LIS}(\sigma_{n})\geq\operatorname{const}\times|\sigma_{n}|^{1/2}$.
We recall that the Erdös-Szekeres theorem [ES35] gives
$\operatorname{LIS}(\sigma)\times\operatorname{LDS}(\sigma)\geq n$ for every
$\sigma\in\mathcal{S}_{n}$.

######  Problem 7.2.

For $n\in\mathbbm{N}$, let $\sigma_{n}$ be a uniform Baxter, semi-Baxter,
strong-Baxter, or meandric permutation of size $n$. Show that there exists
$\alpha>1/2$ (depending only on the type of permutation, not on $n$) such that
$\operatorname{LIS}(\sigma_{n})\geq n^{\alpha}$ with high probability when $n$
is large.

Problem 7.2 is closely related to the following SLE/LQG problem.

######  Problem 7.3.

Assume that we are in the setting of Section 1.2. What is the maximal
Hausdorff dimension (with respect to the LQG metric $D_{h}$) of a set
$X\subset\mathbbm{C}$ which is hit in the same order by $\eta_{1}$ and
$\eta_{2}$?

If $\Delta$ is the maximal Hausdorff dimension in Problem 7.2 and $d_{\gamma}$
is the Hausdorff dimension of $\mathbbm{C}$ w.r.t. $D_{h}$, then it is natural
to conjecture that
$\operatorname{LIS}(\sigma_{n})\approx|\sigma_{n}|^{\Delta/d_{\gamma}}$ for
sufficiently “nice” sequences of permutations
$\\{\sigma_{n}\\}_{n\in\mathbbm{N}}$ which converge in law to
${\boldsymbol{\pi}}$.

The following question is motivated by the formula for pattern densities in
Theorem 1.22. We are especially interested in the case when $\kappa=8$, since
this case corresponds to the meandric permuton.

######  Problem 7.4.

Let $\kappa>4$ and let $\eta$ be a whole-plane space-filling SLEκ from
$\infty$ to $\infty$. Also let $z_{1},\dots,z_{n}\in\mathbbm{C}$ be distinct
points. Is there an exact formula for the function

$P_{\kappa}(z_{1},\dots,z_{n})=\mathbbm{P}\mathopen{}\mathclose{{}\left[\text{$\eta$
hits $z_{j}$ before $z_{j+1}$, $\forall j=1,\dots,n-1$}}\right]?$ (7.1)

Recall Conjecture 1.8, which asserts that a uniform _cyclic_ meandric
permutation converges to the identity permuton. We explained in Section 6.6
why this conjecture should follow from Conjecture 1.3.

######  Problem 7.5.

Is there an unconditional proof of Conjecture 1.8 which does not use
Conjecture 1.3?

The techniques and many of the results of this paper apply to a wide range of
different random permutons which can be described in terms of SLE and LQG. It
is therefore natural to ask the following.

######  Problem 7.6.

Are there any other interesting random permutons, besides the meandric
permuton and the skew Brownian permutons, which can be described by SLE-
decorated LQG in the manner of Section 1.2?

The physics background provided in Section 6 motivates several conjectures
about scaling limits. We highlight the following problem, which is motivated
by the arguments of Sections 6.4 and 6.5.

######  Problem 7.7.

Prove that the scaling limit of a uniform Eulerian triangulation decorated by
a Hamiltonian cycle on its dual is described by LQG with matter central charge
${\mathbf{c}_{\mathrm{M}}}=-1$ (i.e.,
$\gamma=\sqrt{\frac{2}{3}(7-\sqrt{13})}$) decorated by an SLE8 curve.
Moreover, prove the other scaling limit conjectures from Section 6.5 (see also
Conjecture 6.2 and the two bottom boxes in the table in Figure 11).

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