# Central limit theorem for components in meandric systems through high
moments

Svante Janson  Department of Mathematics, Uppsala University, PO Box 480,
SE-751 06 Uppsala, Sweden<EMAIL_ADDRESS>http://www.math.uu.se/$\sim$svante/ and  Paul Thévenin  Institut für
Mathematik, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
<EMAIL_ADDRESS>
(Date: 3 March, 2023)

###### Abstract.

We investigate here the behaviour of a large typical meandric system, proving
a central limit theorem for the number of components of given shape. Our main
tool is a theorem of Gao and Wormald, that allows us to deduce a central limit
theorem from the asymptotics of large moments of our quantities of interest.

###### 2020 Mathematics Subject Classification:

60C05

Supported by the Knut and Alice Wallenberg Foundation

##  1\. Model and main result

###  1.1. Definitions and some notation

Let $n\geq 1$ be an integer. A _meandric system_ of size $n$ is a collection
of non-crossing loops in the plane that intersect the horizontal axis exactly
at the points $[2n]:=\\{1,\ldots,2n\\}$, we call these points the _vertices_
of the meandric system; two meandric systems that differ only by a continuous
deformation of the plane that fixes the horizontal axis are regarded as the
same. Meandric systems were introduced, to our knowledge, by Di Francesco,
Golinelli and Guitter [1], and have recently become again a topic of interest
[6, 4, 2]. A meandric system can be regarded as a set of $n$ non-crossing arcs
with endpoints $[2n]$ in the upper half-plane, and another such set in the
lower half-plane; a meandric system thus determines two non-crossing matchings
(pair-partitions) of $[2n]$, one for each half-plane, and it is easily seen
that this yields a bijection between meandric systems of size $n$ and pairs of
two non-crossing matchings of $[2n]$. In particular, since the number of non-
crossing matchings of $[2n]$ is the Catalan number

(1.1) $\displaystyle\operatorname{Cat}_{n}:=\frac{(2n)!}{n!\,(n+1)!},$

see e.g. [7, item 61], the number of meandric systems of size $n$ is
$\operatorname{Cat}_{n}^{2}$.

Each connected component of a meandric system is a single loop, intersecting
the horizontal axis in a subset of $[2n]$, say
$\left\\{i_{1}<\dots<i_{2k}\right\\}$, which we call the _support_ of the
loop. Note that necessarily there is an even number of vertices in the
support, and an even number of integers in each gap $(i_{j},i_{j+1})$, i.e.,
$i_{j+1}-i_{j}$ is odd for $1\leq j<2k$. We say that two such loops have the
same shape if they differ only by a translation. Thus we may normalize each
shape to have leftmost vertex $1$, and make the following formal definition:

######  Definition 1.1.

A _shape_ is a (connected) non-crossing loop $S$ with support a set of
integers $\left\\{i_{1}=1<i_{2}<\cdots<i_{2k}=2\ell\right\\}$, for some
$k,\ell\geq 1$, such that $i_{j+1}-i_{j}$ is odd for all $1\leq j\leq 2k-1$.

Let $M$ be a meandric system and $C$ a connected component of $M$. We say that
$C$ has shape $S$ if $C$ and $S$ differ only by a translation.

Our main theorem in the following. We prove two special cases as Theorems 3.1
and 4.4, and prove the remaining, more difficult, case in Section 4.2.

######  Theorem 1.2.

Fix a shape $S$. Let $M_{n}$ be a uniformly random meandric system of size $n$
(that is, on $\llbracket 1,2n\rrbracket$) and denote by $X_{S,n}$ the number
of connected components of $M_{n}$ with shape $S$. Then, $X_{S,n}$ satisfies a
central limit theorem: there exist $\mu_{S},\sigma_{S}>0$ such that

(1.2)
$\displaystyle\frac{X_{S,n}-n\mu_{S}}{\sigma_{S}\sqrt{n}}\underset{n\rightarrow\infty}{\overset{(d)}{\longrightarrow}}\mathcal{N}(0,1),$

where $\mathcal{N}(0,1)$ denotes the standard normal distribution.

Observe that the convergence
$\frac{X_{S,n}}{n}\underset{n\rightarrow\infty}{\overset{\mathbb{P}}{\rightarrow}}\mu_{S}$
for some constant $\mu_{S}$ was already obtained in [4], with an explicit
expression for $\mu_{S}$.

##  2\. Preliminaries

###  2.1. More notation

For integers $m\leq n$, $\llbracket m,n\rrbracket$ denotes the integer
interval $[m,n]\cap\mathbb{Z}$. The _size_ of $\llbracket m,n\rrbracket$ is
its number of points, i.e., $n-m+1$. Note that $[n]=\llbracket 1,n\rrbracket$.

For a component $C$ of a meandric system, we denote by $L_{C}$ ($R_{C}$), the
leftmost (rightmost) point in the support of $C$. Furthermore, we say that the
_base_ of $C$ is the interval $\llbracket L_{C},R_{C}\rrbracket$ and let
$\ell(C)$ denote the _half-length_ of $C$, defined as half the size of its
base, i.e., $\ell(C):=\frac{1}{2}(R_{C}-L_{C}+1)$. (Note that $\ell(C)$ always
is an integer.) We use the same definitions for a shape $S$; then $L_{S}=1$
and thus $R_{C}=2\ell(S)$.

For integers $N\geq k\geq 0$, we let

(2.1)
$\displaystyle(N)_{k}:=N(N-1)\cdots(N-k+1)=\frac{N!}{(N-k)!}=k!\,\binom{N}{k},$

the $k$-th descending factorial of $N$.

We use standard $o$ and $O$ notation. Furthermore, for two (positive)
sequences $a_{n}$ and $b_{n}$, $a_{n}\sim b_{n}$ means $a_{n}/b_{n}\to 1$ as
${n\to\infty}$, i.e., $a_{n}=b_{n}(1+o(1))$, and $a_{n}=\Theta(b_{n})$ means
that there exist constants $c>0$ and $C$ such that $c\leq a_{n}/b_{n}\leq C$
for sufficiently large $n$. Note that, for example, $a_{n,r}\sim b_{n,r}$ for
$r=O(\sqrt{n})$ means that this holds for every sequence $r=r(n)=O(\sqrt{n})$,
which is equivalent to $a_{n,r}\sim b_{n,r}$ uniformly for $r\leq C\sqrt{n}$,
for any $C<\infty$; uniformity in $r$ is thus automatic in such cases. We
write “uniformly for $r=O(\sqrt{n})$” for “uniformly for $r\leq C\sqrt{n}$,
for any $C<\infty$”. Unspecified limits are as $n\to\infty$.

###  2.2. The key tool: Gao and Wormald’s theorem.

Our proof relies on a theorem due to Gao and Wormald [5], stating that we can
deduce a central limit theorem for a sequence of variables from the asymptotic
behaviour of their high (factorial) moments. Let us recall this result.

######  Theorem 2.1 (Gao & Wormald [5]).

Let $\mu_{n}s_{n}>-1$ and set $\sigma_{n}:=\sqrt{\mu_{n}+\mu_{n}^{2}s_{n}}$,
where $0<\mu_{n}\rightarrow\infty$. Suppose that $\sigma_{n}=o(\mu_{n})$,
$\mu_{n}=o(\sigma_{n}^{3})$, and that a sequence $\\{X_{n}\\}$ of nonnegative
random variables satisfies as $n\rightarrow\infty$:

(2.2)
$\displaystyle\operatorname{\mathbb{E}}{}[(X_{n})_{r}]\sim\mu_{n}^{r}\exp\left(\frac{r^{2}s_{n}}{2}\right).$

uniformly for all integers $r$ in the range $c\mu_{n}/\sigma_{n}\leq k\leq
C\mu_{n}/\sigma_{n}$, for some constants $C>c>0$. Then
$(X_{n}-\mu_{n})/\sigma_{n}$ converges in distribution to the standard normal
as $n\rightarrow\infty$.

In other words, if high factorial moments of a variable asymptotically match
those of a normal distribution, then convergence to the normal distribution
holds.

###  2.3. Some lemmas

We state some simple lemmas that will be used later. The first is a well known
estimate that we often will use in the sequel.

######  Lemma 2.2.

  1. (i)

If $0\leq k\leq n/2$, then

(2.3)
$\displaystyle(n)_{k}=n^{k}\exp\Bigl{(}-\frac{k^{2}}{2n}+O\Bigl{(}\frac{k^{3}}{n^{2}}+\frac{k}{n}\Bigr{)}\Bigr{)}.$

  2. (ii)

In particular, if $k=O(\sqrt{n})$, then

(2.4)
$\displaystyle(n)_{k}=n^{k}\exp\Bigl{(}-\frac{k^{2}}{2n}+o(1)\Bigr{)}\sim
n^{k}\exp\Bigl{(}-\frac{k^{2}}{2n}\Bigr{)}.$

  3. (iii)

More generally, if $0\leq k\leq m$ with $m=O(\sqrt{n})$, then

(2.5) $\displaystyle(n-m+k)_{k}\sim
n^{k}\exp\Bigl{(}-\frac{m^{2}-(m-k)^{2}}{2n}\Bigr{)}=n^{k}\exp\Bigl{(}-\frac{k(2m-k)}{2n}\Bigr{)}.$

###### Proof.

(i), (ii): This follows easily from a Taylor expansion of $\log(1-i/n)$ for
$0\leq i<k$; we omit the details.

(iii): This follows from (ii) and $(n-m+k)_{k}=(n)_{m}/(n)_{m-k}$. ∎

As one consequence, we obtain the following asymptotics.

######  Lemma 2.3.

Let $n\to\infty$ and $0\leq r=O(\sqrt{n})$. Then

(2.6)
$\displaystyle\frac{\operatorname{Cat}_{n-r}}{\operatorname{Cat}_{n}}\sim
2^{-2r}.$

###### Proof.

The definition (1.1) and Lemma 2.2 yield

(2.7)
$\displaystyle\frac{\operatorname{Cat}_{n-r}}{\operatorname{Cat}_{n}}=\frac{(n)_{r}(n+1)_{r}}{(2n)_{2r}}\sim\frac{(n)_{r}^{2}}{(2n)_{2r}}=\frac{n^{2r}}{(2n)^{2r}}\exp\Bigl{(}-2\frac{r^{2}}{2n}+\frac{(2r)^{2}}{4n}+o(1)\Bigr{)}\sim
2^{-2r}.$

∎

We end this section with another elementary and well known result.

######  Lemma 2.4.

Let $m,n,k\geq 1$. The number of unordered $k$-tuples of disjoint intervals of
size $m$ in $[n]$ is given by

(2.8) $\displaystyle\binom{n-k(m-1)}{k}.$

###### Proof.

By deleting all points except the leftmost in each chosen interval, we obtain
a bijection between the set of such $k$-tuples of intervals and the set of
$k$-tuples of distinct points in $[n-k(m-1)]$. ∎

##  3\. A first example: components of half-length $1$.

As a warm-up, we consider first the simple case where $S$ is the loop of half-
length $1$. For any $i\in[2n]$, we let $Y_{i}$ be the indicator that the
following holds:

$i$$i+1$

Then,

(3.1) $\displaystyle X_{S,n}=\sum_{i=1}^{2n-1}Y_{i}$

and thus, for every $r\geq 1$, summing over $1\leq i_{1}<\dots<i_{r}<2n$,

(3.2) $\displaystyle\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}]$
$\displaystyle=\operatorname{\mathbb{E}}{}\biggl{[}r!\sum_{i_{1}<\dots<i_{r}}Y_{i_{1}}\dotsm
Y_{i_{r}}\biggr{]}=r!\sum_{i_{1}<\dots<i_{r}}\operatorname{\mathbb{E}}{}\bigl{[}Y_{i_{1}}\dotsm
Y_{i_{r}}\bigr{]}.$

The expectation in the last sum is non-zero if and only if the $r$
subintervals $\llbracket i_{j},i_{j}+1\rrbracket$ of $\llbracket
1,2n\rrbracket$ are disjoint, so by Lemma 2.4 there are $\binom{2n-r}{r}$ non-
zero terms. Each of the non-zero terms is $1/\operatorname{Cat}_{n}^{2}$ times
the number of meandric systems of size $n$ that contain $r$ given loops of
half-length $1$; by deleting these loops (and the vertices in them), we obtain
a bijection between such meandric systems and the meandric systems of size
$n-r$, and hence the number of them is $\operatorname{Cat}_{n-r}^{2}$.
Consequently, (3.2) yields

(3.3) $\displaystyle\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}]$
$\displaystyle=(2n-r)_{r}\frac{\operatorname{Cat}_{n-r}^{2}}{\operatorname{Cat}_{n}^{2}}=\frac{(2n)_{2r}}{(2n)_{r}}\cdot\frac{\operatorname{Cat}_{n-r}^{2}}{\operatorname{Cat}_{n}^{2}}.$

In particular, using Lemmas 2.2 and 2.3, if $r=O(\sqrt{n})$, then

(3.4)
$\displaystyle\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}]\sim(2n)^{2r-r}\exp\Bigl{(}-\frac{4r^{2}}{4n}+\frac{r^{2}}{4n}\Bigr{)}2^{-4r}=\left(\frac{n}{8}\right)^{r}\exp\left(-\frac{3r^{2}}{4n}\right).$

In other words, (2.2) holds (uniformly) for $0\leq r\leq C\sqrt{n}$, for any
fixed $C<\infty$, with

(3.5) $\displaystyle\mu_{n}$ $\displaystyle:=\frac{n}{8},$ (3.6)
$\displaystyle s_{n}$ $\displaystyle:=-\frac{3}{2n}.$

We have $\mu_{n}s_{n}=-3/16>-1$, and thus

(3.7)
$\displaystyle\sigma_{n}:=\sqrt{\mu_{n}(1+\mu_{n}s_{n})}=\sqrt{\frac{13}{128}n}.$

We thus have $\sigma_{n}=o(\mu_{n})$ and $\mu_{n}=o(\sigma_{n}^{3})$, and
consequently Theorem 2.1 applies and yields:

######  Theorem 3.1.

If $S$ is a simple loop of half-length 1, then

(3.8)
$\displaystyle\frac{X_{S,n}-n/8}{\sqrt{13n/128}}\underset{n\rightarrow\infty}{\overset{(d)}{\longrightarrow}}\mathcal{N}(0,1).$

This is Theorem 1.2 for this particular choice of $S$, with $\mu_{S}=1/8$ and
$\sigma^{2}_{S}=13/128$.

##  4\. Extension to any fixed shape

Let us now show how we can extend this result to any fixed shape $S$. We now
let $Y_{i}$ be the indicator that there is a component $C$ of shape $S$ such
that $L_{C}=i$; note that (3.1) and (3.2) still hold.

Recall that $\ell(S)$ is the half-length of $S$, so $S$ has base $\llbracket
1,2\ell(S)\rrbracket$. We also define here three other constants
$K(S),c_{+}(S),c_{-}(S)$ depending on $S$. To avoid heavy notation, we will
drop the argument $S$ in what follows, and only denote them by
$K,c_{+},c_{-}$.

######  Definition 4.1.

(See an example in Figure 1.) Observe that a component $C$ of shape $S$, taken
along with the horizontal axis, splits the plane into two unbounded faces,
each belonging to one of the half-planes, and a certain number of bounded
faces. Let $F_{+}$ denote the unbounded face in the upper half-plane, $F_{-}$
the one in the lower half-plane, and $\mathcal{F}(C)$ the set of bounded
faces. For a face $F$, let $\nu(F)$ be the number of vertices in $\llbracket
L_{C},R_{C}\rrbracket$ that lie on the boundary of $F$ but _not_ on $C$, and
observe that necessarily $\nu(F)$ is even. We then set

(4.1) $\displaystyle K(S)$
$\displaystyle:=\prod_{F\in\mathcal{F}(C)}\operatorname{Cat}_{\nu(F)/2},$
(4.2) $\displaystyle c_{+}(S)$ $\displaystyle:=\nu(F_{+})/2,$ (4.3)
$\displaystyle c_{-}(S)$ $\displaystyle:=\nu(F_{-})/2.$

Note that these constants do not depend on the set of vertices on which $C$ is
defined, but only on its shape $S$.

$F_{+}$$F_{-}$$F_{3}$$F_{1}$$F_{2}$$F_{4}$ Figure 1. A component $C$ with four
bounded faces $F_{1},F_{2},F_{3},F_{4}$. In this example, we have
$K(S)=\operatorname{Cat}_{1}^{2}\operatorname{Cat}_{2}\operatorname{Cat}_{3}=10$,
$c_{+}(S)=1$ and $c_{-}(S)=0$, where $S$ is the shape of $C$.

###  4.1. Strong shapes

We say that two components _overlap_ if their bases overlap. Hence, if the
components have the same shape $S$, and the leftmost points in their supports
are $i$ and $j$, they overlap if $|j-i|<2\ell(S)$.

For simplicity, we study first the case when this cannot happen. We say that a
shape $S$ is _strong_ if two different components of a meandric system that
both have shape $S$ cannot overlap. Thus, if $S$ is strong, then
$Y_{i}Y_{j}=0$ when $|j-i|<2\ell(S)$. The simple loop in Section 3 and the
loop in Figure 1 are examples of strong shapes. A shape that is not strong is
called _weak_ ; an example is given in Figure 2.

######  Proposition 4.2.

Let $S$ be a strong shape of half-length $\ell(S)$. Then, for all $r\geq 1$,
we have

(4.4)
$\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}]=\bigl{(}2n-2r\ell(S)+r\bigr{)}_{r}\,K^{r}\,\frac{\operatorname{Cat}_{n-r\ell(S)+rc_{+}}\operatorname{Cat}_{n-r\ell(S)+rc_{-}}}{\operatorname{Cat}_{n}^{2}}.$

###### Proof.

We argue as in Section 3. As noted above, (3.2) still holds, and since $S$ is
strong, we have $Y_{i}Y_{j}=0$ when $|j-i|<2\ell(S)$. Hence, the number of
non-zero terms in (3.2) is $\binom{2n-r(2\ell(S)-1)}{r}$ by Lemma 2.4. Again,
all non-zero terms have the same value, which is
$1/\operatorname{Cat}_{n}^{2}$ times the number of ways that $r$ given
disjoint loops of shape $S$ can be completed to a meandric system of size $n$.
We can fill in the bounded faces of each component in $K$ ways, and there are
$2n-2r\ell(S)+2rc_{\pm}$ vertices left in the upper and lower components,
respectively, so they may be filled in in
$\operatorname{Cat}_{n-r\ell(S)+rc_{\pm}}$ ways. This yields (4.4). ∎

By Lemmas 2.2 and 2.3, it follows from (4.4) that, (uniformly) for
$r=O(\sqrt{n})$, we have

(4.5)
$\displaystyle\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}]\underset{n\rightarrow\infty}{\sim}\left(\frac{2nK}{4^{2\ell(S)-c_{+}-c_{-}}}\right)^{r}\exp\left(-\frac{r^{2}}{4n}\left[(2\ell(S))^{2}-(2\ell(S)-1)^{2}\right]\right).$

This is (2.2) with

(4.6) $\displaystyle\mu_{n}$
$\displaystyle:=\frac{2nK}{4^{2\ell(S)-c_{+}-c_{-}}},$ (4.7) $\displaystyle
s_{n}$
$\displaystyle:=-\frac{(2\ell(S))^{2}-(2\ell(S)-1)^{2})}{2n}=-\frac{4\ell(S)-1}{2n}.$

In order to apply Theorem 2.1, we need to check that $\mu_{n}s_{n}>-1$, which
boils down to the following.

######  Lemma 4.3.

We have

(4.8) $K(4\ell(S)-1)<4^{2\ell(S)-c_{+}-c_{-}}.$

###### Proof.

Observe that we can bound $K$ using the fact that
$\operatorname{Cat}_{n}\leq\frac{4^{n}}{n+1}$ for all $n$: It is easy to see
that for given $c_{\pm}$, $K$ is largest if there is only one bounded face in
each half-plane, and thus,

(4.9) $\displaystyle
K\leq\operatorname{Cat}_{\ell(S)-c_{+}-1}\operatorname{Cat}_{\ell(S)-c_{-}-1}\leq\frac{4^{2\ell(S)-c_{+}-c_{-}-2}}{(\ell(S)-c_{+})(\ell(S)-c_{-})}\leq\frac{4^{2\ell(S)-c_{+}-c_{-}-2}}{\ell(S)},$

since $c_{+}+c_{-}\leq\ell(S)-1$ (to see this, observe that a vertex cannot
belong to both unbounded faces of $S$, and that at least two vertices belong
to $C$). This yields (4.8) directly. ∎

It is clear that $\mu_{n}\rightarrow\infty$. Furthermore, we have just proved
that $1+\mu_{n}s_{n}$ is a positive constant. Thus
$\sigma_{n}=\Theta(\sqrt{\mu_{n}})$, and hence $\sigma_{n}=o(\mu_{n})$ and
$\mu_{n}=o(\sigma_{n}^{3})$. We can therefore apply Theorem 2.1 to obtain the
central limit theorem in this case too:

######  Theorem 4.4.

Let $S$ be a strong shape. Then

(4.10)
$\displaystyle\frac{X_{S,n}-n\mu_{S}}{\sigma_{S}\sqrt{n}}\underset{n\longrightarrow\infty}{\overset{(d)}{\rightarrow}}\mathcal{N}(0,1),$

where

(4.11)
$\displaystyle\mu_{S}=\frac{2K}{4^{2\ell(S)-c_{+}-c_{-}}}\quad\text{and}\quad\sigma_{S}=\sqrt{\frac{2K}{4^{2\ell(S)-c_{+}-c_{-}}}\Bigl{(}1-\frac{K(4\ell(S)-1)}{4^{2\ell(S)-c_{+}-c_{-}}}\Bigr{)}}.$

This proves Theorem 1.2 in the case when $S$ is a strong shape, with explicit
formulas for $\mu_{S}$ and $\sigma_{S}$.

###  4.2. Weak shapes

Finally, we study the case of a weak shape $S$. Thus, now there may be
overlaps between two components of shape $S$, that is, two indices $i<j$ such
that $|j-i|<2\ell(S)$ and $Y_{i}Y_{j}=1$, where $Y_{i}$ is defined as before.
See Figure 2 for an example.

Figure 2. Two components of same shape overlapping. Here,
$\operatorname{\mathbb{E}}{}[Y_{1}Y_{7}]>0$, while $2\ell(S)=10$.

Let $A^{r}$ be the set of all $r$-tuples $E:=\\{i_{1},\ldots,i_{r}\\}$ with
$1\leq i_{1}<\cdots<i_{r}\leq 2n$, For any such $r$-tuple $E$, define an
equivalence relation $\sim_{E}$ on $\\{1,\ldots,r\\}$ as the smallest one (for
the inclusion of the equivalence classes) satisfying: for all $1\leq
k_{1},k_{2}\leq r$ such that $|i_{k_{1}}-i_{k_{2}}|<2\ell(S)$,
$k_{1}\sim_{E}k_{2}$. We call the equivalence classes of $\sim_{E}$ _blocks_.
Furthermore, for $1\leq j\leq r$, we let $A^{r}_{j}$ be the set of $r$-tuples
$E\in A^{r}$ that have exactly $j$ blocks. Thus
$A^{r}=\bigcup_{j=1}^{r}A^{r}_{j}$. Note that $A^{r}_{r}$ is the set of
$r$-tuples $E$ such that all blocks are singletons. An $r$-tuple $E$
corresponds to a collection $(C_{k})_{1}^{r}$ of loops of shape $S$, shifted
such that $C_{k}$ has $L_{C_{k}}=i_{k}$. In particular, $E\in A_{r}^{r}$ if
and only if these loops are non-overlapping.

Define, for all $1\leq u\leq r$:

(4.12) $\displaystyle
F_{u}:=\binom{2n-2u\ell(S)+u}{u}\,K^{u}\,\frac{\operatorname{Cat}_{n-u\ell(S)+uc_{+}}\operatorname{Cat}_{n-u\ell(S)+uc_{-}}}{\operatorname{Cat}_{n}^{2}}.$

By the argument in the proof of Proposition 4.2, $u!\,F_{u}$ is the
contribution to $\operatorname{\mathbb{E}}{}[(X_{S,n})_{u}]$ from $u$-tuples
of non-overlapping components.

We have the following estimates:

######  Lemma 4.5.

Let $S$ be a weak shape.

  * (i)

For all $r\geq 1$,

(4.13) $\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}]\geq r!F_{r}.$

  * (ii)

For all $1\leq u\leq r$,

(4.14) $\displaystyle\sum_{E\in
A^{r}_{u}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in E}Y_{i}\right]$
$\displaystyle\leq\binom{r-1}{u-1}(2\ell(S))^{r-u}F_{u}.$

  * (iii)

For each fixed $M\geq 0$, uniformly for $r=O(\sqrt{n})$ with $r\geq 2M$,

(4.15) $\displaystyle\sum_{E\in
A^{r}_{r-M}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]=\Theta\left(r^{M}F_{r-M}\right)$

and, if also $r\to\infty$,

(4.16) $\displaystyle\sum_{E\in
A^{r}_{r-M}(1,2)}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]=(1-o(1))\sum_{E\in
A^{r}_{r-M}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in E}Y_{i}\right],$

where $A^{r}_{r-M}(1,2)$ is the subset of $A^{r}_{r-M}$ made only of blocks of
sizes $1$ or $2$.

###### Proof of Lemma 4.5.

(i): We rewrite (3.2) as

(4.17)
$\displaystyle\operatorname{\mathbb{E}}{}\left[(X_{S,n})_{r}\right]=r!\sum_{E\in
A^{r}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]=r!\sum_{u=1}^{r}\sum_{E\in
A^{r}_{u}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in E}Y_{i}\right].$

The term with $u=r$ yields the contribution from $r$-tuples of non-overlapping
components, which as noted after (4.12) is $r!F_{r}$.

(ii): For each $r$-tuple $E\in A^{r}_{u}$, keep in the product only the
leftmost point of each block, observing that, for any sets $A\subseteq
B\subseteq\llbracket 1,2n\rrbracket$, we have
$\operatorname{\mathbb{E}}{}\left[\prod_{i\in
B}Y_{i}\right]\leq\operatorname{\mathbb{E}}{}\left[\prod_{i\in
A}Y_{i}\right]$. Note that this set of leftmost points belongs to $A^{u}_{u}$.
If the size of the $i$-th leftmost block is $j_{i}$, then for each set of
leftmost points, the number of possible positions of the other $j_{i}-1$
points in the block is at most $(2\ell(S))^{j_{i}-1}$, since each point after
the first is within $2\ell(S)$ of the preceding one. Hence,

(4.18) $\displaystyle\sum_{E\in
A^{r}_{u}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in E}Y_{i}\right]$
$\displaystyle\leq\,\sum_{\begin{subarray}{c}j_{1}+\ldots+j_{u}=r\\\
j_{1},\ldots,j_{u}\geq
1\end{subarray}}{\prod_{i=1}^{u}(2\ell(S))^{j_{i}-1}}\cdot\sum_{E^{\prime}\in
A^{u}_{u}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E^{\prime}}Y_{i}\right]=\,\sum_{\begin{subarray}{c}j_{1}+\ldots+j_{u}=r\\\
j_{1},\ldots,j_{u}\geq
1\end{subarray}}{\prod_{i=1}^{u}(2\ell(S))^{j_{i}-1}}\cdot F_{u}.$

Finally, this yields (4.14), since the number of allowed sequences
$(j_{1},\dots,j_{u})$ is $\binom{r-1}{u-1}$, and
$\prod_{i=1}^{u}(2\ell(S))^{j_{i}-1}=(2\ell(S))^{r-u}$ for all of them.

(iii): We partition the set $A^{r}_{r-M}$ as follows. Consider an
$(r-M)$-tuple $T:=(T_{1},\ldots,T_{r-M})$ of integers $\geq 1$, of sum $r$,
and consider also a function $J$ which, to each $1\leq i\leq r-M$, associates
a $T_{i}$-tuple $J_{i}$ of integers $1=:j_{i,1}<j_{i,2}<\ldots<j_{i,T_{i}}$
such that, for all $1\leq k\leq T_{i}-1$, $j_{i,k+1}-j_{i,k}<2\ell(S)$, and,
furthermore, the $T_{i}$ loops of shape $S$ that start at the vertices $j_{k}$
($k=1,\dots,T_{i}$) are disjoint so that they may occur together as components
in a meandric system. (We call such pairs $(T,J)$ _admissible_.) Denote by
$A_{T,J}$ the subset of $A^{r}_{r-M}$ made of $r$-tuples $E$ such that the
$i$-th leftmost block of $E$ has size $T_{i}$, and if this block is
$\left\\{a^{1}_{i},\ldots,a^{T_{i}}_{i}\right\\}$, then we have
$a_{i}^{k+1}-a_{i}^{k}=j_{i,k+1}-j_{i,k}$ for all $1\leq k\leq T_{i}-1$. In
other words, $A_{T,J}$ accounts for all $r$-tuples of components with $r-M$
blocks, where the sizes of the blocks are given, as well as the intervals
between the starting points of each component of shape $S$ in each block.
Hence, $A^{r}_{r-M}$ is the union $\bigcup A_{T,J}$ over all admissible pairs
$(T,J)$.

Since we only consider $(r-M)$-tuples $T$ such that

(4.19) $\displaystyle r=\sum_{i=1}^{r-M}T_{i}=r-M+\sum_{i=1}^{r-M}(T_{i}-1),$

there at most $M$ indices $i$ with $T_{i}>1$, and thus at least $r-2M$ indices
with $T_{i}=1$. Note also that if $T_{i}=1$, then trivially $J_{i}=(1)$. Given
an admissible pair $(T,J)$, we define the reduced pair
$(\widehat{T},\widehat{J})$ by deleting all $T_{i}$ and $J_{i}$ such that
$T_{i}=1$ from $T$ and $J$; thus $\widehat{T}:=(T_{i}:1\leq i\leq r-M\text{
and }T_{i}>1)$ and similarly for $\widehat{J}$. Consequently, $\widehat{T}$
and $\widehat{J}$ are both sequences of (the same) length $\leq M$. Since
(4.19) implies that their entries are bounded (for a fixed $M$), there is only
a finite set $\mathcal{T}$ of reduced pairs $(\widehat{T},\widehat{J})$, where
$\mathcal{T}$ depends on $M$ and $S$ but not on $r$.

Conversely, given an admissible reduced pair $(\widehat{T},\widehat{J})$, with
$\widehat{T}=(\widehat{T}_{1},\dots,\widehat{T}_{k})$, we can obtain
$(\widehat{T},\widehat{J})$ from $\binom{r-M}{k}$ different (admissible) pairs
$(T,J)$. Note that here, by (4.19), since each $\widehat{T}_{i}\geq 2$,

(4.20) $\displaystyle
k\leq\sum_{i=1}^{k}(\widehat{T}_{i}-1)=\sum_{i=1}^{r-M}(T_{i}-1)=M,$

with equality if and only if $\widehat{T}_{i}=2$ for all $i\leq k$.

We now want to understand the behaviour of $\sum_{E\in
A_{T,J}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in E}Y_{i}\right]$ for an
admissible pair $(T,J)$. In a way similar to Proposition 4.2 (using an
extension of Lemma 2.4 to intervals of different lengths), we obtain

(4.21) $\displaystyle\sum_{E\in
A_{T,J}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]=\binom{2n-2\widetilde{\ell}+(r-M)}{r-M}\,\widetilde{K}\,\frac{\operatorname{Cat}_{n-d_{+}}\operatorname{Cat}_{n-d_{-}}}{\operatorname{Cat}_{n}^{2}},$

where, for any $E\in A_{T,J}$, $\widetilde{\ell}$ is the sum of the half-
lengths of the blocks, $\widetilde{K}$ accounts for the bounded faces defined
by the horizontal axis and the loops defined by $E$, and $d_{+},d_{-}$ for the
unbounded faces. (Note that these constants are the same for all $E\in
A_{T,J}$ so they depend only on $T$ and $J$.) Moreover, since at least $r-2M$
of these blocks are singletons, and the remaining blocks are determined by
$\widehat{T}$ and $\widehat{J}$, we can write

(4.22) $\displaystyle\widetilde{K}=K^{r-2M}K^{\prime},$

for some $K^{\prime}>0$ depending only on $(\widehat{T},\widehat{J})$.
Similarly,

(4.23) $\displaystyle\widetilde{\ell}$
$\displaystyle=(r-M)\ell(S)+\ell^{\prime},$ (4.24) $\displaystyle d_{+}$
$\displaystyle=(r-M)(\ell(S)-c_{+})+e_{+},$ (4.25) $\displaystyle d_{-}$
$\displaystyle=(r-M)(\ell(S)-c_{-})+e_{-}$

for some $\ell^{\prime},e_{+},e_{-}$ depending only on
$(\widehat{T},\widehat{J})$. In particular, for a fixed $M$, it follows that
$K^{\prime},\ell^{\prime},e_{+},e_{-}$ can only take a fixed number of values
independently of $n$ and $r$.

We compare (4.21) and $F_{r-M}$ given by (4.12). First, by Lemma 2.2(iii),

(4.26)
$\displaystyle\frac{\binom{2n-2\widetilde{\ell}+(r-M)}{r-M}}{\binom{2n-2(r-M)\ell(S)+(r-M)}{r-M}}=\frac{\bigl{(}2n-2\widetilde{\ell}+(r-M)\bigr{)}_{r-M}}{\bigl{(}2n-2(r-M)\ell(S)+(r-M)\bigr{)}_{r-M}}$
$\displaystyle\hskip
40.00006pt\sim\exp\Bigl{(}-\frac{r-M}{4n}\bigl{(}(4\widetilde{\ell}-r+M)-(4(r-M)\ell(S)-r+M)\bigr{)}\Bigr{)}=\exp\bigl{(}o(1)\bigr{)},$

since $\widetilde{\ell}=r\ell(S)+O(1)$ by (4.23) and $r=o(n)$. Similarly, as a
consequence of Lemma 2.3 and (4.24)–(4.25),

(4.27)
$\displaystyle\frac{\operatorname{Cat}_{n-d_{\pm}}}{\operatorname{Cat}_{n-(r-M)(\ell(S)-c_{\pm})}}\sim
4^{-d_{\pm}+(r-M)(\ell(S)-c_{\pm})}=4^{-e_{\pm}}.$

Consequently, using also (4.22), we obtain from (4.21) and (4.12),

(4.28) $\displaystyle\frac{\sum_{E\in
A_{T,J}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]}{F_{r-M}}=C_{T,J}(1+o(1)),$

where $C_{T,J}>0$ only depends on $(\widehat{T},\widehat{J})$, and therefore
only takes a finite number of values. In particular,

(4.29) $\displaystyle\sum_{E\in
A_{T,J}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]=\Theta\bigl{(}F_{r-M}\bigr{)},$

and this holds uniformly for $r=O(\sqrt{n})$ and all admissible $(T,J)$.

By (4.20) and the discussion before it, there are
$\binom{r-M}{k}=\Theta(r^{k})$ admissible pairs $(T,J)$ for each
$(\widehat{T},\widehat{J})$, where $k\leq M$, with equality when all
$\widehat{T}_{i}=2$. Note that since we assume that the shape $S$ is weak,
there exists at least one such admissible $(\widehat{T},\widehat{J})$ with
$\widehat{T}=(2,\dots,2)$. Hence, summing (4.29) over all $(T,J)$ yields
(4.15).

Moreover, $A^{r}_{r-M}\setminus A^{r}_{r-M}(1,2)$ is the union
$\bigcup^{\prime}A_{T,J}$ where we only sum over admissible pairs $(T,J)$ with
some $T_{i}\geq 3$; these correspond to reduced pairs
$(\widehat{T},\widehat{J})$ with some $\widehat{T}_{i}\geq 3$, and we see from
(4.20) that each such reduced pair has length $\leq M-1$, and thus corresponds
to $O(r^{M-1})$ admissible pairs. Consequently, summing (4.29) over all
$(T,J)$ of this type yields

(4.30) $\displaystyle\sum_{E\in A^{r}_{r-M}\setminus
A^{r}_{r-M}(1,2)}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]=O\bigl{(}r^{M-1}F_{r-M}\bigr{)}=o\bigl{(}r^{M}F_{r-M}\bigr{)},$

which yields (4.16) by (4.15). ∎

The next proposition shows that, in order to get the asymptotic behaviour of
$\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}]$, we only need to take into
account the configurations whose number of blocks that are not singletons is a
given constant.

######  Proposition 4.6.

Fix a weak shape $S$. Then, there exists $\eta>0$ such that, for any
$\varepsilon>0$, there exists $M>0$ such that we have, uniformly for
$r\leq\eta\sqrt{n}$,

(4.31) $\displaystyle\sum_{u\leq r-M}\sum_{E\in
A^{r}_{u}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]\leq\varepsilon
F_{r}\leq\varepsilon\frac{1}{r!}\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}].$

######  Remark 4.7.

For convenience, we assume here that $r/\sqrt{n}$ is small. In fact,
Proposition 4.6 can easily be extended to $r\leq C\sqrt{n}$ for any $C$ (with
$M$ depending on $C$ and $\varepsilon$), but we have no need for this.

To prove this, we start with a lemma:

######  Lemma 4.8.

There exists $Q>0$ depending only on the shape $S$ such that, for $n$ large
enough, for all $u\leq\sqrt{n}$:

(4.32) $\displaystyle\frac{F_{u+1}}{F_{u}}\geq Q\frac{n}{u}.$

###### Proof.

We just compute the ratio term by term, recalling (4.12). We have
$\frac{K^{u+1}}{K^{u}}=K$. The ratio of the ratios of Catalan numbers
converges uniformly to a positive constant. Finally, the ratio of binomial
coefficients is, using Lemma 2.2,

(4.33)
$\displaystyle\frac{u!}{(u+1)!}\cdot\frac{\bigl{(}2n-(u+1)(2\ell(S)-1)\bigr{)}_{u+1}}{\bigl{(}2n-u(2\ell(S)-1)\bigr{)}_{u}}=\frac{1}{u+1}\cdot\frac{(2n)^{u+1}}{(2n)^{u}}\exp\bigl{(}O(1)\bigr{)}\geq
c\frac{n}{u}$

for some $c>0$ and all large $n$ and $u\leq\sqrt{n}$. The result follows. ∎

###### Proof of Proposition 4.6.

Using Lemma 4.5(ii), we have for all $M\geq 0$:

(4.34) $\displaystyle\sum_{u\leq r-M}\sum_{E\in
A^{r}_{u}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in E}Y_{i}\right]$
$\displaystyle\leq\sum_{u=1}^{r-M}\binom{r-1}{u-1}(2\ell(S))^{r-u}F_{u}.$

Letting

(4.35) $\displaystyle B_{r,u}:=\binom{r-1}{u-1}(2\ell(S))^{r-u}F_{u},$

we get from Lemma 4.8 that, for $r\leq\sqrt{n}$ and any $u\leq r-1$:

(4.36)
$\displaystyle\frac{B_{r,u+1}}{B_{r,u}}=\frac{1}{2\ell(S)}\frac{r-u}{u}\frac{F_{u+1}}{F_{u}}\geq\frac{Q}{2\ell(S)}\frac{n(r-u)}{u^{2}}\geq\frac{Q}{2\ell(S)}\frac{n}{u^{2}}.$

Hence, there exists $\eta>0$ small enough such that, for all
$u<r\leq\eta\sqrt{n}$, we have $B_{r,u+1}\geq 2B_{r,u}$, and thus by backward
induction,

(4.37) $\displaystyle B_{r,u}\leq 2^{-(r-u)}B_{r,r}.$

Then, for $r\leq\eta\sqrt{n}$, (4.34) yields

(4.38) $\displaystyle\sum_{u\leq r-M}\sum_{E\in
A^{r}_{u}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in E}Y_{i}\right]$
$\displaystyle\leq\sum_{u=1}^{r-M}B_{r,u}\leq 2^{1-M}B_{r,r}=2^{1-M}F_{r}.$

This yields (4.31) if we choose $M$ such that $2^{1-M}\leq\varepsilon$, since
$r!\,F_{r}\leq\operatorname{\mathbb{E}}{}[(X_{S,n})_{r}]$ by the comment after
(4.12). ∎

Proposition 4.6 shows that we only need to understand the asymptotic behaviour
of the configurations with a number of blocks $r-M$ for given $M\geq 0$, and
Lemma 4.5(iii) that we can focus on configurations with blocks of size $1$ or
$2$. To actually prove our final result, we need to refine Lemma 4.5(iii) and
obtain the explicit constants that appear. We define another set of constants,
which will account for the cases with blocks of size 2, i.e., cases when two
components of shape $S$ overlap.

######  Definition 4.9.

Let $S$ be a shape. There is a finite set of integers $i\geq 1$ such that
$\operatorname{\mathbb{E}}{}[Y_{1}Y_{i}]>0$ and $i-1<2\ell(S)$. Let $I(S)$ be
this set, and $i_{1},\ldots,i_{k}$ its elements. For $i\in I(S)$, let
$\ell_{i}$, $K_{i}$, $c_{+}(i)$ and $c_{-}(i)$ be the equivalents of
$\ell(S),K,c_{+},c_{-}$ in this case of two components $C,C^{\prime}$ that
overlap and start at positions $1$ and $i$. In particular,
$\ell_{i}=\ell(S)+(i-1)/2$ is the total half-length of the block made of two
components of shape $S$ started at positions $1$ and $i$. Furthermore, $C$ and
$C^{\prime}$ together with the horizontal axis define two unbounded faces
($F_{+}$ in the upper half-plane and $F_{-}$ in the lower half-plane), and
several bounded faces; let $\mathcal{F}(C,C^{\prime})$ be the set of bounded
faces. For each face $F$, let $\nu(F)$ be the number of integers in
$\llbracket L(C),R(C)\rrbracket\cup\llbracket
L(C^{\prime}),R(C^{\prime})\rrbracket=\llbracket L(C),R(C^{\prime})\rrbracket$
that are incident to $F$ but do not belong to $C$ nor to $C^{\prime}$. We set
$K_{i}:=\prod_{F\in\mathcal{F}(C,C^{\prime})}\operatorname{Cat}_{\nu(F)/2}$.
Finally, we define $c_{\pm}(i):=\nu(F_{\pm})/2$. Observe again that all these
constants only depend on $S$ and $i$.

Note that $i\in I(S)$ may be even; in this case $2\ell_{i}$, $\nu(F_{+})$ and
$\nu(F_{-})$ are odd, and thus $\ell_{i}$ and $c_{\pm}(i)$ are half-integers.

######  Lemma 4.10.

Let $r=O(\sqrt{n})$ with $r\to\infty$. Then, for every fixed $M\geq 0$,

(4.39) $\displaystyle\sum_{E\in
A^{r}_{r-M}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]\underset{n\rightarrow\infty}{\sim}F_{r}\sum_{\begin{subarray}{c}g_{i}\geq
0,i\in I(S)\\\ \sum_{i}g_{i}=M\end{subarray}}\prod_{i\in
I(S)}\frac{\bigl{(}b_{i}\frac{r^{2}}{2n}\bigr{)}^{g_{i}}}{g_{i}!},$

where

(4.40) $\displaystyle b_{i}$
$\displaystyle:=4^{4\ell(S)-2\ell_{i}+c_{+}(i)-2c_{+}+c_{-}(i)-2c_{-}}\frac{K_{i}}{K^{2}}.$

Note that $b_{i}$ measures (in a specific way) how much two overlapping
components of shape $S$ differ from two non-overlapping ones.

###### Proof.

For each $I(S)$-tuple $G=(g_{i})_{i\in I(S)}$ of integers with sum $M$, let
$A^{r}_{r-M,G}$ be the set of $r$-tuples $1\leq i_{1}<\ldots<i_{r}\leq 2n$
with $r-2M$ blocks of size $1$ and $M$ blocks of size $2$, such that for each
$i\in I(S)$, there are $g_{i}$ blocks of type
$\left\\{i_{k},i_{k+1}=i_{k}+i-1\right\\}$ with $k<r$. Then $A^{r}_{r-M,G}$ is
the union of some classes $A_{T,J}$ from the proof of Lemma 4.5, with all
$T_{i}\in\left\\{1,2\right\\}$ and a specified number $g_{i}$ of $k$ such that
$J_{k}=(1,i)$. Hence, we obtain from (4.21), where the multinomial coefficient
is the number of $(T,J)$ that are included in $A^{r}_{r-M,G}$,

(4.41) $\displaystyle\sum_{E\in
A^{r}_{r-M,G}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]=\binom{r-M}{g_{i_{1}},\ldots,g_{i_{k}},r-2M}\binom{2n-2\widetilde{\ell}+(r-M)}{r-M}\,\widetilde{K}\,\frac{\operatorname{Cat}_{n-d_{+}}\operatorname{Cat}_{n-d_{-}}}{\operatorname{Cat}_{n}^{2}},$

where, by (4.22)–(4.25) and the argument yielding them:

(4.42) $\displaystyle\widetilde{K}$ $\displaystyle=K^{r-2M}\prod_{i\in
I(S)}K_{i}^{g_{i}},$ (4.43) $\displaystyle\widetilde{\ell}$
$\displaystyle=(r-2M)\ell(S)+\sum_{i\in I(S)}g_{i}\ell_{i},$ (4.44)
$\displaystyle d_{\pm}$
$\displaystyle=\widetilde{\ell}-(r-2M)c_{\pm}-\sum_{i\in
I(S)}g_{i}c_{\pm}(i).$

We now argue similarly as in the proof of Lemma 4.5, but this time we compare
to $F_{r}$. We have

(4.45) $\displaystyle\binom{r-M}{g_{1},\ldots,g_{k},r-2M}\sim r^{M}\prod_{i\in
I(S)}\frac{1}{g_{i}!},$ (4.46)
$\displaystyle\frac{\binom{2n-2\widetilde{\ell}+(r-M)}{r-M}}{\binom{2n-2r\ell(S)+r}{r}}=\frac{r!}{(r-M)!}\cdot\frac{\bigl{(}2n-2\widetilde{\ell}+(r-M)\bigr{)}_{r-M}}{\bigl{(}2n-2r\ell(S)+r\bigr{)}_{r}}$
$\displaystyle\hskip 20.00003pt\sim
r^{M}(2n)^{-M}\exp\Bigl{(}-\frac{1}{4n}\bigl{(}(r-M)(4\widetilde{\ell}-r+M)-r(4r\ell(S)-r)\bigr{)}\Bigr{)}\sim
r^{M}(2n)^{-M},$ (4.47)
$\displaystyle\frac{\widetilde{K}}{K^{r}}=K^{-2M}\prod_{i\in
I(S)}K_{i}^{g_{i}},$ (4.48)
$\displaystyle\frac{\operatorname{Cat}_{n-d_{\pm}}}{\operatorname{Cat}_{n-r\ell(S)+rc_{\pm}}}\sim
4^{-d_{\pm}+r(\ell(S)-c_{\pm})}=4^{2M(\ell(S)-c_{\pm})-\sum_{i\in
I(S)}(\ell_{i}-c_{\pm}(i))g_{i}}.$

and thus, from (4.41) and (4.12), recalling that $\sum_{i\in I(S)}g_{i}=M$,

(4.49) $\displaystyle\frac{\sum_{E\in
A^{r}_{r-M,G}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]}{F_{r}}$
$\displaystyle\qquad\underset{n\rightarrow\infty}{\sim}r^{2M}(2nK^{2})^{-M}4^{2(\ell(S)-c_{+})M-\sum_{i\in
I(S)}(\ell_{i}-c_{+}(i))g_{i}}4^{2(\ell(S)-c_{-})M-\sum_{i\in
I(S)}(\ell_{i}-c_{-}(i))g_{i}}\prod_{i\in I(S)}\frac{1}{g_{i}!}K_{i}^{g_{i}}$
$\displaystyle\qquad=\left(B\frac{r^{2}}{2n}\right)^{M}\prod_{i\in
I(S)}\frac{q_{i}^{g_{i}}}{g_{i}!}=\prod_{i\in
I(S)}\frac{\bigl{(}Bq_{i}\frac{r^{2}}{2n}\bigr{)}^{g_{i}}}{g_{i}!},$

where

(4.50) $\displaystyle B$
$\displaystyle:=\frac{4^{4\ell(S)-2c_{+}-2c_{-}}}{K^{2}},$ (4.51)
$\displaystyle q_{i}$ $\displaystyle:=4^{-2\ell_{i}+c_{+}(i)+c_{-}(i)}K_{i}.$

The set $A^{r}_{r-M}(1,2)$ defined in Lemma 4.5(iii) is the union of
$A^{r}_{r-M,G}$ over all $G$ with sum $M$. Hence, (4.49) implies, noting that
there is only a finite number of such $G$,

(4.52) $\displaystyle\sum_{E\in
A^{r}_{r-M}(1,2)}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]\underset{n\rightarrow\infty}{\sim}F_{r}\sum_{\begin{subarray}{c}g_{i}\geq
0,i\in I(S)\\\ \sum_{i}g_{i}=M\end{subarray}}\prod_{i\in
I(S)}\frac{\bigl{(}Bq_{i}\frac{r^{2}}{2n}\bigr{)}^{g_{i}}}{g_{i}!}.$

The result (4.39) now follows from (4.52) and (4.16), using $Bq_{i}=b_{i}$. ∎

###### Proof of Theorem 1.2 for weak shapes.

Let $r\to\infty$ with $r\leq\eta\sqrt{n}$, where $\eta$ is as in Proposition
4.6.

We may sum (4.39) over all $M\geq 0$ (with $A^{r}_{r-M}:=\emptyset$ for
$M>r$), since Proposition 4.6 shows that we may approximate the sum by a
finite sum with a fixed number of terms. Consequently, recalling (4.17),

(4.53) $\displaystyle\operatorname{\mathbb{E}}{}\left[(X_{S,n})_{r}\right]$
$\displaystyle=r!\sum_{M=0}^{\infty}\sum_{E\in
A^{r}_{r-M}}\operatorname{\mathbb{E}}{}\left[\prod_{i\in
E}Y_{i}\right]{\sim}r!\,F_{r}\sum_{M=0}^{\infty}\sum_{\begin{subarray}{c}g_{i}\geq
0,i\in I(S)\\\ \sum_{i}g_{i}=M\end{subarray}}\prod_{i\in
I(S)}\frac{\bigl{(}b_{i}\frac{r^{2}}{2n}\bigr{)}^{g_{i}}}{g_{i}!}$
$\displaystyle=r!\,F_{r}\prod_{i\in
I(S)}\exp\Bigl{(}b_{i}\frac{r^{2}}{2n}\Bigr{)}.$

By Lemmas 2.2 and 2.3, (4.12) implies (similarly to (4.5))

(4.54) $\displaystyle
r!\,F_{r}\sim\left(\frac{2nK}{4^{2\ell(S)-c_{+}-c_{-}}}\right)^{r}\exp\Bigl{(}-\frac{r^{2}}{4n}\bigl{(}4\ell(S)-1\bigr{)}\Bigr{)}.$

Finally, (4.53) and (4.54) yield, for $r\to\infty$ with $r\leq\eta\sqrt{n}$,

(4.55)
$\displaystyle\operatorname{\mathbb{E}}{}\left[(X_{S,n})_{r}\right]\underset{n\rightarrow\infty}{\sim}\left(\frac{2nK}{4^{2\ell(S)-c_{+}-c_{-}}}\right)^{r}\exp\left(-\frac{r^{2}}{4n}\left(4\ell(S)-1\right)+\frac{r^{2}}{2n}\sum_{i\in
I(S)}b_{i}\right).$

This is (2.2), with

(4.56) $\displaystyle\mu_{n}$
$\displaystyle:=\frac{2nK}{4^{2\ell(S)-c_{+}-c_{-}}},$ (4.57) $\displaystyle
s_{n}$ $\displaystyle:=\frac{-(4\ell(S)-1)+2\sum_{i\in I(S)}b_{i}}{2n}.$

In particular, (2.2) thus holds for $r=r(n)$ with $\frac{\eta}{2}\sqrt{n}\leq
r\leq\eta\sqrt{n}$; as noted in Section 2.1, it then automatically holds
uniformly in this range. Furthermore,

(4.58)
$\displaystyle\mu_{n}s_{n}\geq-\frac{K(4\ell(S)-1)}{4^{2\ell(S)-c_{+}-c_{-}}}>-1$

by Lemma 4.3, and we have again $\mu_{n}=\Theta(n)$ and
$\sigma_{n}=\Theta(\sqrt{n})$. It follows that Theorem 2.1 applies in this
case too, which yields (1.2). ∎

We obtain from (4.56)–(4.57)

(4.59)
$\displaystyle\sigma^{2}_{S}=\frac{2K}{4^{2\ell(S)-c_{+}-c_{-}}}\Bigl{(}1+\frac{K}{4^{2\ell(S)-c_{+}-c_{-}}}\Bigl{(}1-4\ell(S)+2\sum_{i\in
I(S)}b_{i}\Bigr{)}\Bigr{)},$

with $b_{i}$ given by (4.40). Note that this formula holds also for strong
shapes (when $I(S)=\emptyset$) by (4.11).

##  5\. Open problems

We list here some open problems concerning possible extensions of our results.

  1. 1.

It seems possible to extend the arguments above to joint factorial moments

(5.1)
$\displaystyle\operatorname{\mathbb{E}}{}\bigl{[}(X_{S_{1},n})_{r_{1}}\dotsm(X_{S_{k},n})_{r_{k}}\bigr{]}$

for several shapes $S_{1},\dots,S_{k}$, and then obtain a multivariate version
of Theorem 1.2 using a multivariate version of Gao and Wormald’s theorem [3],
[8]. However, we have not checked the details. Such a multivariate theorem
would immediately imply, for example, a central limit theorem for the number
of components of a given half-length.

  2. 2.

Considering shapes that are similar, can we obtain a central limit theorem for
the number of components that only cross the horizontal axis twice (i.e., the
support has size 2, but the half-length is arbitrary)?

  3. 3.

Is is true, as Kargin [6] has conjectured, that the total number of components
is asymptotically normal?

## References

  * [1] P. Di Francesco, O. Golinelli and E. Guitter. Meanders and the Temperley-Lieb algebra. _Commun. Math. Phys._ 186 (1997), 1–-59.
  * [2] Jacopo Borga, Ewain Gwynne and Minjae Park. On the geometry of uniform meandric systems. https://arxiv.org/abs/2212.00534.
  * [3] Gabriel Berzunza Ojeda, Cecilia Holmgren and Svante Janson. Fringe trees for random trees with given vertex degrees. In preparation.
  * [4] Valentin Féray and Paul Thévenin. Components in meandric systems and the infinite noodle. _Int. Math. Res. Not. IMRN_ (2022), rnac156.
  * [5] Jason Gao and Nicholas Wormald. Asymptotic normality determined by high moments, and submap counts of random maps. _Probab. Theory Related Fields_ 130 (2004), 368–376.
  * [6] Vladislav Kargin. Cycles in random meander systems. _J. Stat. Phys._ 181 (2020), no. 6, 2322–2345.
  * [7] Richard P. Stanley. _Catalan numbers_. Cambridge University Press, New York, 2015.
  * [8] Nick Wormald. Personal communication.


a subset $Y\in{\bf L}(U)$ and a function
$\xi:Y^{c}\to\ell^{2}(Y^{c})_{1,+},x\mapsto\xi_{x}$ satisfying:

  1. (1)

$\mathrm{supp}(\xi_{x})\subseteq B(x,S)$ for any $x\in Y^{c}$;

  2. (2)

for any $x,y\in Y^{c}$ with $d(x,y)\leq R$, we have
$\|\xi_{x}-\xi_{y}\|_{1}\leq\varepsilon$.

Similar to the proof in the case of Property A, we also have the following:

######  Corollary 8.8.

Let $(X,d)$ be a space, $U\subseteq\beta X$ an invariant open subset and ${\bf
L}={\bf L}(U)$ the associated ideal in $X$. Then $X$ has partial Property A
towards $U^{c}$ _if and only if_ for any $R>0$ and $\varepsilon>0$, there
exist $S>0$, a subset $Y\in{\bf L}(U)$ and a kernel $k:Y^{c}\times
Y^{c}\to\mathbb{R}$ of positive type satisfying:

  1. (1)

for $x,y\in Y^{c}$, we have $k(x,y)=k(y,x)$ and $k(x,x)=1$;

  2. (2)

for $x,y\in Y^{c}$ with $d(x,y)\geq S$, we have $k(x,y)=0$;

  3. (3)

for $x,y\in Y^{c}$ with $d(x,y)\leq R$, we have $|1-k(x,y)|\leq\varepsilon$.

###  8.2. Partial operator norm localisation property

Recall that the notion of operator norm localisation property (ONL) was
introduced by Chen, Tessera, Wang and Yu in [10], and proved by Sako in [37]
that ONL is equivalent to Property A. Here we introduce a partial version of
ONL, parallel to Definition 8.1.

Let $\nu$ be a positive locally finite Borel measure on $X$ and $\mathcal{H}$
be a separable infinite-dimensional Hilbert space. For an operator
$T\in\mathfrak{B}(L^{2}(X,\nu)\otimes\mathcal{H})$, we can also define its
propagation as in Section 2.3. We introduce the following:

######  Definition 8.9.

Let $(X,d)$ be a space and $U\subseteq\beta X$ an invariant open subset. We
say that $X$ has _partial operator norm localisation property (partial ONL)
towards $U^{c}=\partial_{\beta}X\setminus U$_ if there exists $c\in(0,1]$ such
that for any $R>0$ there exist $S>0$ and $D\subseteq X$ with
$\overline{D}\supseteq U^{c}$ satisfying the following: for any positive
locally finite Borel measure $\nu$ on $X$ with $\mathrm{supp}(\nu)\subseteq D$
and any $a\in\mathfrak{B}(L^{2}(X,\nu)\otimes\mathcal{H})$ with propagation at
most $R$, there exists a non-zero $\zeta\in L^{2}(X,\nu)\otimes\mathcal{H}$
with $\mathrm{diam}(\mathrm{supp}(\zeta))\leq S$ such that
$c\|a\|\cdot\|\zeta\|\leq\|a\zeta\|$.

The aim of the rest of this subsection is to show that partial ONL is
equivalent to partial Property A. We will follow the outline of [37].

To simplify the statement, denote
$\mathbb{C}^{R}_{u}[X;\mathcal{H}]:=\\{T\in\mathfrak{B}(\ell^{2}(X)\otimes\mathcal{H}):\mathrm{prop}(a)\leq
R\\}$ for $R\geq 0$. For a subspace $Y\subseteq X$, it is clear that
$\mathbb{C}^{R}_{u}[Y]\cong\chi_{Y}\mathbb{C}^{R}_{u}[X]\chi_{Y}$ (resp.
$\mathbb{C}^{R}_{u}[Y;\mathcal{H}]\cong\chi_{Y}\mathbb{C}^{R}_{u}[X;\mathcal{H}]\chi_{Y}$),
and hence can be regarded as a subset of $\mathbb{C}^{R}_{u}[X]$ (resp.
$\mathbb{C}^{R}_{u}[X;\mathcal{H}]$) with support in $Y\times Y$. Similarly,
$C^{*}_{u}(Y)\cong\chi_{Y}C^{*}_{u}(X)\chi_{Y}$ can be regarded as a
$C^{*}$-subalgebra in $C^{*}_{u}(X)$. Hence we will not tell the difference in
the sequel. For $S>0$, denote

$B^{Y}_{S}:=\prod_{x\in X}\mathfrak{B}(\ell^{2}(B(x,S)\cap Y)),$

whose elements will be written as $b=([b^{x}(y,z)]_{y,z\in B(x,S)\cap
Y})_{x\in X}$. We also consider the map

$\psi_{S}^{Y}:C^{*}_{u}(Y)\cong\chi_{Y}C^{*}_{u}(X)\chi_{Y}\longrightarrow
B^{Y}_{S}\quad\text{by}\quad a\mapsto([a(y,z)]_{y,z\in B(x,S)\cap Y})_{x\in
X}.$

Recall the notions of completely positive map and completely bounded map:

  * •

A self-adjoint closed subspace $F$ of a unital $C^{*}$-algebra $B$ such that
$1_{B}\in F$ is called an _operator system_.

  * •

A linear map $\phi$ from $F$ to a $C^{*}$-algebra $C$ is said to be
_completely positive_ if the map $\phi^{(n)}=\phi\otimes\mathrm{id}:F\otimes
M_{n}(\mathbb{C})\to C\otimes M_{n}(\mathbb{C})$ is positive for every $n$.

  * •

A linear map $\theta:F\to C$ is said to be _completely bounded_ if the
sequence $\\{\|\theta^{(n)}:F\otimes M_{n}(\mathbb{C})\to C\otimes
M_{n}(\mathbb{C})\|\\}$ is bounded. Denote
$\|\theta\|_{\mathrm{cb}}:=\sup_{n\in\mathbb{N}}\|\theta^{(n)}\|$.

We have the following characterisation for partial ONL, which is analogous to
[37, Proposition 3.1]. The proof is almost identical, hence omitted.

######  Lemma 8.10.

Let $(X,d)$ be a space and $U\subseteq\beta X$ an invariant open subset. Then
the following are equivalent:

  1. (1)

$X$ has partial ONL towards $U^{c}$;

  2. (2)

there exists $c\in(0,1]$ such that for any $R>0$ there exist $S>0$ and
$D\subseteq X$ with $\overline{D}\supseteq U^{c}$ satisfying condition
$(\alpha)$: for any $a\in\mathbb{C}^{R}_{u}[D;\mathcal{H}]$ there exists a
non-zero $\zeta\in\ell^{2}(X)\otimes\mathcal{H}$ with
$\mathrm{diam}(\mathrm{supp}(\zeta))\leq S$ and
$c\|a\|\cdot\|\zeta\|\leq\|a\zeta\|$;

  3. (3)

for any $c\in(0,1)$ and $R>0$, there exist $S>0$ and $D\subseteq X$ with
$\overline{D}\supseteq U^{c}$ satisfying condition $(\alpha)$;

  4. (4)

for any $c\in(0,1)$ and $R>0$, there exist $S>0$ and $D\subseteq X$ with
$\overline{D}\supseteq U^{c}$ satisfying condition $(\beta)$: for any
$a\in\mathbb{C}^{R}_{u}[D]$ there exists a non-zero $\xi\in\ell^{2}(X)$ with
$\mathrm{diam}(\mathrm{supp}(\xi))\leq S$ and
$c\|a\|\cdot\|\xi\|\leq\|a\xi\|$;

  5. (5)

for any $\varepsilon,R>0$ there exist $S>R$ and $D\subseteq X$ with
$\overline{D}\supseteq U^{c}$ such that

$\|(\psi_{S}^{D}|_{\mathbb{C}^{R}_{u}[D]})^{-1}:\psi_{S}^{D}(\mathbb{C}^{R}_{u}[D])\longrightarrow\mathbb{C}^{R}_{u}[D]\|<1+\varepsilon;$

  6. (6)

for any $\varepsilon,R>0$ there exist $S>R$ and $D\subseteq X$ with
$\overline{D}\supseteq U^{c}$ such that

$\|(\psi_{S}^{D}|_{\mathbb{C}^{R}_{u}[D]})^{-1}:\psi_{S}^{D}(\mathbb{C}^{R}_{u}[D])\longrightarrow\mathbb{C}^{R}_{u}[D]\|_{\mathrm{cb}}<1+\varepsilon;$

We record the following, which comes directly from Lemma 8.7 and 8.10.

######  Lemma 8.11.

Let $(X,d)$ be a space, $U\subseteq\beta X$ an invariant open subset and ${\bf
L}={\bf L}(U)$ the associated ideal in $X$. Then $X$ has partial ONL towards
$U^{c}$ _if and only if_ for any $c\in(0,1)$ and $R>0$ there exist $S>0$ and
$Y\in{\bf L}(U)$ satisfying the following: for any
$a\in\mathbb{C}^{R}_{u}[Y^{c}]$ there exists a non-zero $\xi\in\ell^{2}(X)$
with $\mathrm{diam}(\mathrm{supp}(\xi))\leq S$ and
$c\|a\|\cdot\|\xi\|\leq\|a\xi\|$.

Finally, we can mimic the proof of [37, Theorem 4.1] using Proposition 8.2′,
Remark 8.6, Corollary 8.8 and Lemma 8.10 instead, and reach the following. The
proof is almost identical, and hence omitted.

######  Proposition 8.12.

Let $(X,d)$ be a space and $U\subseteq\beta X$ be an invariant open subset.
Then $X$ has partial Property A towards $U^{c}$ _if and only if_ $X$ has
partial ONL towards $U^{c}$.

To end this subsection, we study a permanence property of partial ONL, which
will help to prove the main result in the next subsection.

Let $X$ be a space and ${\bf L}$ an ideal in $X$. Assume that $X$ can be
decomposed into $X=X_{1}\cup X_{2}$. Consider

(8.3) ${\bf L}_{i}:=\\{Y\cap X_{i}:Y\in{\bf L}\\}\quad\text{for}\quad i=1,2.$

Then it is routine to check that ${\bf L}_{i}$ is an ideal in $X_{i}$ for
$i=1,2$, and

${\bf L}=\\{Y_{1}\cup Y_{2}:Y_{i}\in{\bf L}_{i},i=1,2\\}.$

With respect to the decomposition above, we now show that partial ONL is
preserved under finite unions.

######  Proposition 8.13.

With the notation as above, assume that $X_{i}$ has partial ONL towards $\beta
X_{i}\setminus U({\bf L}_{i})$ for $i=1,2$. Then $X$ has partial ONL towards
$\beta X\setminus U({\bf L})$.

One way to prove Proposition 8.13 is to follow the proof of [16, Lemma 3.3]
with minor changes. Here we choose another approach using Proposition 8.12.

Recall from Corollary A.8 that for a subset $Z\subseteq X$, the closure
$\overline{Z}$ in $\beta X$ is homeomorphic to $\beta Z$. Hence we will regard
them as the same object in the sequel. Now we can easily transfer the
restriction of ideals in (8.3) to that of invariant open subsets (the proof is
straightforward, hence omitted):

######  Lemma 8.14.

Let $X=X_{1}\cup X_{2}$, ${\bf L}$ be an ideal in $X$ and $U=U({\bf
L})\subseteq\beta X$ be the associated invariant open subset of $\beta X$.
Then $U\cap\overline{X_{i}}$ is an invariant open subset of
$\overline{X_{i}}=\beta X_{i}$, which corresponds to the ideal ${\bf L}_{i}$
in (8.3) for $i=1,2$.

Although $U\cap\overline{X_{i}}$ is invariant in $\beta X_{i}$, generally it
is not invariant in $\beta X$. This coincides with the fact that
$\chi_{X_{i}}C^{*}_{u}(X)\chi_{X_{i}}\cong C^{*}_{u}(X_{i})$ is a just
subalgebra in $C^{*}_{u}(X)$ rather than an ideal. Instead, we consider the
spatial ideal $I_{X_{i}}$ recalled in Section 4, and prove the following
permanence property for partial Property A:

######  Lemma 8.15.

Let $X=X_{1}\cup X_{2}$, $U\subseteq\beta X$ be an invariant open subset and
$U_{i}=U\cap\overline{X_{i}}$ for $i=1,2$. If $X_{i}$ has partial Property A
towards $\beta X_{i}\setminus U_{i}$ for $i=1,2$, then $X$ has partial
Property A towards $\beta X\setminus U$.

###### Proof.

For any $R>0$ and $i=1,2$, set $U_{i}(R):=\overline{\bigcup_{Y\in{\bf
L}(U_{i})}\mathcal{N}_{R}(Y)}$. Clearly, $U_{i}(R)$ is an invariant open
subset of $\overline{\mathcal{N}_{R}(X_{i})}=\beta(\mathcal{N}_{R}(X_{i}))$.
Moreover, it follows from Proposition 8.2 that $\mathcal{N}_{R}(X_{i})$ has
partial Property A towards $\overline{\mathcal{N}_{R}(X_{i})}\setminus
U_{i}(R)$. By definition, this means that the groupoid
$G(\mathcal{N}_{R}(X_{i}))_{\overline{\mathcal{N}_{R}(X_{i})}\setminus
U_{i}(R)}$ is amenable. Hence as a subgroupoid,
$G(\mathcal{N}_{R}(X_{i}))_{\overline{\mathcal{N}_{R}(X_{i})}\setminus U}$ is
amenable, which further implies that the groupoid
$\bigcup_{R>0}G(\mathcal{N}_{R}(X_{i}))_{\overline{\mathcal{N}_{R}(X_{i})}\setminus
U}$ is amenable.

Note from Lemma 4.14 that
$\bigcup_{R}\overline{\mathcal{N}_{R}(X_{i})}\setminus U$ is invariant in
$\beta X$ and Lemma 4.16 implies that

$G(X)_{\bigcup_{R}\overline{\mathcal{N}_{R}(X_{i})}\setminus
U}=\bigcup_{R>0}G(\mathcal{N}_{R}(X_{i}))_{\overline{\mathcal{N}_{R}(X_{i})}\setminus
U},$

which is hence amenable. Note that
$\bigcup_{R}\overline{\mathcal{N}_{R}(X_{1})}\cup\bigcup_{R}\overline{\mathcal{N}_{R}(X_{2})}=\beta
X$, and hence due to the extension property we obtain that

$G(X)_{\beta X\setminus
U}=G(X)_{\bigcup_{R}\overline{\mathcal{N}_{R}(X_{1})}\setminus U}\cup
G(X)_{\bigcup_{R}\overline{\mathcal{N}_{R}(X_{2})}\setminus U}$

is amenable as required. ∎

Combining Proposition 8.12 and Lemma 8.15, we conclude Proposition 8.13.

###  8.3. Characterisation for $I(U)=\tilde{I}(U)$

Having established all the necessary ingredients above, now we present the
main result of this section:

######  Theorem 8.16.

Let $(X,d)$ be a space as in Section 2.2 and $U\subseteq\beta X$ be a
countably generated invariant open subset. Then the following are equivalent:

  1. (1)

$X$ has partial Property A towards $\beta X\setminus U$;

  2. (2)

$\tilde{I}(U)=I(U)$;

  3. (3)

the ideal $I_{G}$ of all ghost operators is contained in $I(U)$.

Note that $U=X$ is countably generated, and hence Theorem 8.16 recovers [36,
Theorem 1.3] (see Example 4.8 and Example 5.5). Borrowing the language of
[36], condition (3) in Theorem 8.16 says that all the ghosts can be busted in
the geometric ideal $I(U)$.

We follow the outline of the proof for [36, Theorem 1.3]. Firstly, we need a
modified version of [36, Lemma 4.2]:

######  Lemma 8.17.

Let $(X,d)$ be a space, $U\subseteq\beta X$ be a countably generated invariant
open subset and ${\bf L}={\bf L}(U)$ the associated ideal in $X$. Assume that
$X$ does not have partial ONL towards $U^{c}$. Then there exist
$\kappa\in(0,1)$, $R>0$, a sequence $(T_{n})$ in $\mathbb{C}_{u}[X]$, a
sequence $(B_{n})$ of finite subsets of $X$ and a sequence $(S_{n})$ of
positive real numbers such that:

  1. (a)

$(S_{n})$ is an increasing sequence tending to infinity as $n\to\infty$;

  2. (b)

each $T_{n}$ is positive and has norm $1$;

  3. (c)

for $n\neq m$, then $B_{n}\cap B_{m}=\emptyset$;

  4. (d)

each $T_{n}$ is supported in $B_{n}\times B_{n}$;

  5. (e)

for each $n$ and $\xi\in\ell^{2}(X)$ with $\|\xi\|=1$ and
$\mathrm{diam}(\mathrm{supp}\xi)\leq S_{n}$, then $\|T_{n}\xi\|\leq\kappa$;

  6. (f)

for each $Y\in{\bf L}(U)$, there exists $n$ such that $B_{n}\cap Y=\emptyset$.

###### Proof.

Fixing a basepoint $x_{0}\in X$, consider the decomposition $X=X^{(1)}\cup
X^{(2)}$ with

$X^{(1)}:=\bigsqcup_{m\text{ even}}\\{x\in X:m^{2}\leq
d(x,x_{0})<(m+1)^{2}\\}$

and

$X^{(2)}:=\bigsqcup_{m\text{ odd}}\\{x\in X:m^{2}\leq
d(x,x_{0})<(m+1)^{2}\\}.$

Set ${\bf L}_{i}:=\\{Y\cap X^{(i)}:Y\in{\bf L}\\}$ for $i=1,2$. By assumption,
$X$ does not have partial Property A towards $U^{c}$. Hence without loss of
generality, we can assume that $X^{(1)}$ does not have partial Property A
towards $\beta X^{(1)}\setminus U({\bf L}_{1})$ thanks to Proposition 8.13.
Note that this implies that $U({\bf L}_{1})\neq\beta X^{(1)}$. It is clear
that ${\bf L}_{1}$ is also countably generated, and hence according to Lemma
7.6 there exists a countable subset $\\{Y_{1},Y_{2},\cdots,Y_{n},\cdots\\}$ of
${\bf L}_{1}$ such that

${\bf L}_{1}=\\{Z\subseteq X^{(1)}:~{}\exists~{}n\in\mathbb{N}\text{ such that
}Z\subseteq Y_{n}\\}.$

In the sequel, we fix such a sequence $\\{Y_{1},Y_{2},\cdots,Y_{n},\cdots\\}$.

Due to Lemma 8.11, we know that there exist $c\in(0,1)$ and $R>0$ such that
for any $Y\in{\bf L}_{1}$ and $S>0$, there exists
$T\in\mathbb{C}^{R}_{u}[X^{(1)}\setminus Y]$ with $\|T\|=1$ satisfying: for
any $\xi\in\ell^{2}(X^{(1)})$ with $\mathrm{diam}(\mathrm{supp}\xi)\leq S$ and
$\|\xi\|=1$, then $\|T\xi\|<c$. We call such an operator _$(R,c,S,Y)$
-localised_. Replacing $T$ by $T^{*}T$ (and $R$ by $2R$ and $c$ by
$\sqrt{c}$), we see that there exist $c\in(0,1)$ and $R>0$ such that for any
$Y\in{\bf L}_{1}$ and $S>0$, there exists a positive $(R,c,S,Y)$-localised
operator of norm one. Let us fix such $c$ and $R$ in the rest of the proof,
and set $\kappa:=\frac{2c}{1+c}<1$.

Note that $X^{(1)}$ can be decomposed into:

$X^{(1)}:=\bigsqcup_{m\in\mathbb{N}}X_{m}$

where each $X_{m}$ is finite and $d(X_{m},X_{n})>R$ for any $n\neq m$. Hence
each $T\in\mathbb{C}^{R}_{u}[X^{(1)}]$ splits as a block diagonal sum of
finite rank operators $T=\bigoplus_{m}T^{(m)}$ where
$T^{(m)}\in\mathfrak{B}(\ell^{2}(X_{m}))$, with respect to this decomposition.

Take $S_{1}=1$. By assumption, there exists a positive
$(R,c,S_{1},Y_{1})$-localised operator
$T\in\mathbb{C}^{R}_{u}[X^{(1)}\setminus Y_{1}]$ with norm $1$. Note that
$\|T\|=\sup_{m}\|T^{(m)}\|$, and then there exists $m_{1}\in\mathbb{N}$ such
that $\|T^{(m_{1})}\|>\frac{1+c}{2}$. We set
$T_{1}:=T^{(m_{1})}/\|T^{(m_{1})}\|$ and denote
$B_{1}:=X_{m_{1}}\cap(X^{(1)}\setminus Y_{1})$, which is nonempty since $T$
has support in $B_{1}\times B_{1}$ by assumption. Then for any
$\xi\in\ell^{2}(X^{(1)})$ with $\|\xi\|=1$ and
$\mathrm{diam}(\mathrm{supp}(\xi))\leq S_{1}$, we have

$\|T_{1}\xi\|\leq\frac{2c}{1+c}=\kappa<1.$

Now we take $S_{2}>\mathrm{max}\big{\\{}\mathrm{diam}\big{(}\bigsqcup_{k\leq
m_{1}}X_{k}\big{)},2\big{\\}}$. By assumption, there exists a positive
$(R,c,S_{2},Y_{2})$-localised operator
$T\in\mathbb{C}^{R}_{u}[X^{(1)}\setminus Y_{2}]$ with norm $1$. Again there
exists $m_{2}$ such that $\|T^{(m_{2})}\|>\frac{1+c}{2}$, which forces
$m_{2}>m_{1}$. We set $T_{2}:=T^{(m_{2})}/\|T^{(m_{2})}\|$ and denote
$B_{2}:=X_{m_{2}}\cap(X^{(1)}\setminus Y_{2})$, which is nonempty since $T$
has support in $B_{2}\times B_{2}$ by assumption. Similarly for any
$\xi\in\ell^{2}(X^{(1)})$ with $\|\xi\|=1$ and
$\mathrm{diam}(\mathrm{supp}(\xi))\leq S_{2}$, we have
$\|T_{2}\xi\|\leq\kappa$.

Inductively, we can construct a sequence $(T_{n})$ in
$\mathbb{C}_{u}[X^{(1)}]\subseteq\mathbb{C}_{u}[X]$, a sequence $(B_{n})$ of
finite subsets of $X^{(1)}\subseteq X$ and a sequence $(S_{n})$ of positive
real numbers satisfying condition (a)-(e) in the statement. Furthermore for
each $Z\in{\bf L}(U)$, there exists $Y_{n}$ containing $Z\cap X^{(1)}$ for
some $n$. By construction, we know that $B_{n}\cap Y_{n}=\emptyset$ and
$B_{n}\subseteq X^{(1)}$. Hence we have:

$B_{n}\cap Z=B_{n}\cap X^{(1)}\cap Z\subseteq B_{n}\cap Y_{n}=\emptyset,$

which provides condition (f) and concludes the proof. ∎

######  Remark 8.18.

Comparing Lemma 8.17 with [36, Lemma 4.2], we note that condition (f) is the
only extra condition added in Lemma 8.17. It seems hard to write condition (f)
in the language of the invariant open subset $U$ instead of the ideal ${\bf
L}(U)$, which indicates the importance of using the notion of ideals in spaces
as mentioned in Section 1.

Now we are in the position to prove Theorem 8.16.

###### Proof of Theorem 8.16.

“(1) $\Rightarrow$ (2)” is contained in Proposition 7.2, and “(2)
$\Rightarrow$ (3)” holds trivially since
$I_{G}=\tilde{I}(X)\subseteq\tilde{I}(U)$. Hence it suffices to show “(3)
$\Rightarrow$ (1)”, and we follow the outline of the proof for [36, Theorem
1.3].

Assume that $X$ does not have partial Property A towards $U^{c}$, then it
follows from Proposition 8.12 that $X$ does not have partial ONL towards
$U^{c}$. Then from Lemma 8.17, there exist $\kappa\in(0,1)$, $R>0$, a sequence
$(T_{n})$ in $\mathbb{C}_{u}[X]$, a sequence $(B_{n})$ of finite subsets of
$X$ and a sequence $(S_{n})$ of positive real numbers satisfying condition
(a)-(f) therein. Now we consider the operator

$T:=\bigoplus_{n}T_{n},$

which is a positive operator in $\mathbb{C}_{u}[X]$ of norm one.

Now we take a continuous function $f:[0,1]\to[0,1]$ such that
$\mathrm{supp}f\subseteq[\frac{1+\kappa}{2},1]$ and $f(1)=1$. Consider the
operator $f(T)\in C^{*}_{u}(X)$, which is positive, norm one, and admits a
decomposition

$f(T)=\bigoplus_{n}f(T_{n}),$

where each $f(T_{n})\in\mathfrak{B}(\ell^{2}(B_{n}))$. We will show that
$f(T)\in\tilde{I}(X)\setminus I(U)$, and hence conclude a contradiction.

First we show that $f(T)\notin I(U)$. Recall from (4.2) that

$I(U)=\overline{\\{T^{\prime}\in\mathbb{C}_{u}[X]:\mathrm{supp}(T^{\prime})\subseteq
Y\times Y\mbox{~{}for~{}some~{}}Y\in{\bf L}(U)\\}},$

Now for any $T^{\prime}\in\mathbb{C}_{u}[X]$ with
$\mathrm{supp}(T^{\prime})\subseteq Y\times Y$ for some $Y\in{\bf L}(U)$,
condition (f) in Lemma 8.17 implies that there exists $n$ such that $B_{n}\cap
Y=\emptyset$. Hence we have:

$\|f(T)-T^{\prime}\|\geq\|\chi_{B_{n}}f(T)\chi_{B_{n}}-\chi_{B_{n}}T^{\prime}\chi_{B_{n}}\|=\|f(T_{n})-0\|=1,$

which implies that $f(T)\notin I(U)$.

On the other hand, using the same argument as for [36, Theorem 1.3] (since
Lemma 8.17 provides all the conditions required in [36, Lemma 4.2]), we obtain
that $f(T)$ is a ghost operator. Hence according to Example 5.5, we have
$f(T)\in\tilde{I}(X)$. Therefore, we conclude the proof. ∎

##  9\. Open questions

Here we collect several open questions around this topic.

First recall from Theorem 5.4 that for a space $(X,d)$, any ideal $I$ in the
uniform Roe algebra $C^{*}_{u}(X)$ must lie between $I(U)$ and $\tilde{I}(U)$
for $U=U(I)$. However, the structure of the lattice

$\mathfrak{I}_{U}=\\{I\mbox{~{}is~{}an~{}ideal~{}in~{}}C^{*}_{u}(X):U(I)=U\\}$

in (1.1) is still unclear. Note that for any invariant open subset $V\supseteq
U$ of $\beta X$, the ideal $I(V)\cap\tilde{I}(U)$ belongs to the lattice
$\mathfrak{I}_{U}$. Unfortunately, we do not know whether these ideals can
bust every element in $\mathfrak{I}_{U}$. Hence we pose the following:

######  Question 9.1.

Let $(X,d)$ be a space and $U\subseteq\beta X$ be an invariant open subset.
Can we describe elements in the lattice
$\mathfrak{I}_{U}=\\{I\mbox{~{}is~{}an~{}ideal~{}in~{}}C^{*}_{u}(X):U(I)=U\\}$
in details? For $I\in\mathfrak{I}_{U}$, can we find an invariant open subset
$V\supseteq U$ such that $I=I(V)\cap\tilde{I}(U)$?

Note that an answer to the above question together with Theorem 5.4 will
provide a full description for the ideal structure of the uniform Roe algebra.

Our next question concerns minimal points discussed in Section 6.1. Recall
that minimal points in the Stone-Čech boundary correspond to maximal ideals in
the uniform Roe algebra. However as shown in Theorem 6.8, there exist a number
of non-minimal points in the boundary. Hence it would be interesting to
explore a practical approach to distinguish minimal points.

######  Question 9.2.

Given a space $(X,d)$, can we find a practical approach to distinguish minimal
points in the Stone-Čech boundary $\partial_{\beta}X$?

Note from Theorem 6.8 that the answer might not be easy even in the elementary
case that $X=\mathbb{Z}$.

Our last questions concern the assumption of countably generatedness used in
Section 7 and 8. Recall that in Proposition 7.8 we prove that the inclusion
$\iota_{U}:I(U)\hookrightarrow\tilde{I}(U)$ induces an isomorphism in
$K$-theory when the space is coarsely embeddable and $U$ is countably
generated. We ask the following:

######  Question 9.3.

Does the inclusion $\iota_{U}:I(U)\hookrightarrow\tilde{I}(U)$ induce an
isomorphism in $K$-theory for coarsely embeddable $X$ without the assumption
that $U$ is countably generated?

Also note that even under the assumption of countable generatedness, we are
merely able to show that
$(\iota_{I})_{\ast}:K_{\ast}(\overline{I\cap\mathbb{C}_{u}[X]})\longrightarrow
K_{\ast}(I)$ is injective in Proposition 7.8. Hence we also pose the
following:

######  Question 9.4.

For an ideal $I$ in $C^{*}_{u}(X)$ with $U(I)$ countably generated and $X$
coarsely embeddable, is the map
$(\iota_{I})_{\ast}:K_{\ast}(\overline{I\cap\mathbb{C}_{u}[X]})\longrightarrow
K_{\ast}(I)$ surjective for $\ast=0,1$?

Our final question is designed for Theorem 8.16. Recall that the assumption of
countably generatedness plays an important role in Lemma 8.17, which is in
turn crucial in the proof of Theorem 8.16. Hence we ask the following:

######  Question 9.5.

Let $(X,d)$ be a space and $U\subseteq\beta X$ be an invariant open subset. If
$\tilde{I}(U)=I(U)$, can we deduce that $X$ has partial Property A towards
$\beta X\setminus U$?

##  Appendix A Ultrafilters

Here we collect some basic knowledge on ultrafilters, which is used throughout
the paper. The material should be fairly well-known (see, _e.g._ , [9,
Appendix A], [34, Chapter 7.4] or [43, Appendix A]). While some of the results
might not be standard and we have not dug into the reference, we include the
proofs for convenience to readers.

######  Definition A.1.

Let $X$ be a set and $\mathcal{P}(X)$ be its power set. An _ultrafilter_ on
$X$ is a family $\mathcal{U}\subseteq\mathcal{P}(X)$ satisfying the following:

  1. (I.1)

$\emptyset\notin\mathcal{U}$;

  2. (I.2)

for $A,B\in\mathcal{U}$, then $A\cap B\in\mathcal{U}$;

  3. (I.3)

for $A\in\mathcal{U}$ and $A\subseteq B$, then $B\in\mathcal{U}$;

  4. (I.4)

for any $A\subseteq X$, either $A\in\mathcal{U}$ or $X\setminus
A\in\mathcal{U}$.

For an ultrafilter $\mathcal{U}$ on $X$, we can associate a function
$\omega:\mathcal{P}(X)\to\\{0,1\\}$ by setting $\omega(A)=1$ if and only if
$A\in\mathcal{U}$. It follows from (I.1)-(I.4) above that $\omega$ is a
finitely additive $\\{0,1\\}$-valued probability measure on $\mathcal{P}(X)$.
Conversely for such a function $\omega$ on $\mathcal{P}(X)$, we can associate
a family $\mathcal{U}_{\omega}:=\\{A\subseteq X:\omega(A)=1\\}$. It is clear
that $\mathcal{U}_{\omega}$ is an ultrafilter on $X$, and these two procedures
are inverse to each other. Therefore throughout the paper, we slide between
these two notions freely without further explanation.

For $a\in X$, it is clear that the family $\\{A\in\mathcal{P}(X):a\in A\\}$ is
an ultrafilter on $X$. Such an ultrafiler is called _principal_. An
ultrafilter which is not principal is called _non-principal_. An argument
using Zorn’s lemma shows that non-principal ultrafilters always exist whenever
$X$ is infinite.

The following is well-known (see, _e.g._ , [43, Lemma A.2]):

######  Lemma A.2.

Let $\omega$ be an ultrafilter on a set $X$, and $D\subseteq X$ with
$\omega(D)=1$. Let $f:D\to Y$ be a function from $D$ to a compact Hausdorff
topological space $Y$. Then there exists a unique point $y\in Y$ such that for
any open neighbourhood $U$ of $y$, we have $\omega(f^{-1}(U))=1$.

######  Definition A.3.

The unique point in Lemma A.2 is called the _ultralimit_ of $f$ along $\omega$
or the _$\omega$ -limit_ of $f$, denoted by $\lim_{\omega}f$ or
$\lim_{a\to\omega}f(a)$.

We record the following localisation result:

######  Lemma A.4.

Let $\mathcal{U}$ be an ultrafilter on a set $X$, and $A\subseteq X$ with
$A\in\mathcal{U}$. Then we have:

  1. (1)

$\\{S\cap A:S\in\mathcal{U}\\}=\\{S\subseteq A:S\in\mathcal{U}\\}$ is an
ultrafilter on $A$, denoted by $\mathcal{U}_{A}$.

  2. (2)

$\mathcal{U}=\\{S\subseteq X:S\cap A\in\mathcal{U}_{A}\\}=\\{S\subseteq
X:\exists~{}S^{\prime}\in\mathcal{U}_{A}\text{ such that }S^{\prime}\subseteq
S\\}$.

###### Proof.

(1). It follows from Definition A.1 that $\\{S\cap
A:S\in\mathcal{U}\\}=\\{S\subseteq A:S\in\mathcal{U}\\}$, and hence
(I.1)-(I.3) hold for $\mathcal{U}_{A}$. Concerning (I.4): given $B\subseteq
A$, if $B\in\mathcal{U}$ then $B\in\mathcal{U}_{A}$ as well; if
$B\notin\mathcal{U}$ then $X\setminus B\in\mathcal{U}$, and hence $A\setminus
B=A\cap(X\setminus B)\in\mathcal{U}_{A}$.

(2). This is straightforward, hence omitted. ∎

We can also extend an ultrafilter on a subset to the whole space. The proof is
straightforward, hence omitted.

######  Lemma A.5.

Let $Y$ be a subset of a set $X$, and $\mathcal{U}_{0}$ an ultrafilter on $Y$.
Define

$\mathcal{U}:=\\{S\subseteq X:S\cap Y\in\mathcal{U}_{0}\\}.$

Then $\mathcal{U}$ is an ultrafilter on $X$.

The following result provides an approach to combine a family of ultrafilters
into a single one:

######  Lemma A.6.

Let $\\{X_{i}\\}_{i\in I}$ be a family of sets, and $\mathcal{U}_{i}$ be an
ultrafilter on $X_{i}$ for each $i\in I$. Let $\omega_{0}$ be an ultrafilter
on $I$. Consider the set $X:=\bigsqcup_{i\in I}X_{i}$ and define:

$\mathcal{U}:=\big{\\{}\bigsqcup_{i\in I}A_{i}\subseteq\bigsqcup_{i\in
I}X_{i}:\exists~{}J\subseteq I\text{ with }\omega_{0}(J)=1\text{ such that
}\forall i\in J,A_{i}\in\mathcal{U}_{i}\big{\\}}.$

Then $\mathcal{U}$ is an ultrafilter on $X$.

###### Proof.

Firstly, it is clear that $\emptyset\notin\mathcal{U}$. Assume that
$\bigsqcup_{i\in I}A_{i}$ and $\bigsqcup_{i\in I}B_{i}\in\mathcal{U}$, _i.e._
, there exist $J_{A},J_{B}\subseteq I$ with
$\omega_{0}(J_{A})=\omega_{0}(J_{B})=1$ such that $A_{i}\in\mathcal{U}_{i}$
for any $i\in J_{A}$ and $B_{i}\in\mathcal{U}_{i}$ for any $i\in J_{B}$.
Consider $(\bigsqcup_{i\in I}A_{i})\cap(\bigsqcup_{i\in
I}B_{i})=\bigsqcup_{i\in I}(A_{i}\cap B_{i})$ and $J=J_{A}\cap J_{B}$. Then
$\omega_{0}(J)=1$ and for each $i\in J$, $A_{i}$ and $B_{i}$ are in
$\mathcal{U}_{i}$. This implies that $A_{i}\cap B_{i}\in\mathcal{U}_{i}$,
which concludes (I.2).

It is clear that (I.3) holds for $\mathcal{U}$ and finally, we consider (I.4).
Given $A=\bigsqcup_{i\in I}A_{i}\subseteq X$, denote $J:=\\{i\in
I:A_{i}\in\mathcal{U}_{i}\\}$. If $\omega_{0}(J)=1$, then it follows that
$A\in\mathcal{U}$. Otherwise, assume that $\omega_{0}(J)=0$. Then we consider
$X\setminus A=\bigsqcup_{i\in I}(X_{i}\setminus A_{i})$. Then $I\setminus
J=\\{i\in I:X_{i}\setminus A_{i}\in\mathcal{U}_{i}\\}$ and
$\omega_{0}(I\setminus J)=1$, which implies that $X\setminus A\in\mathcal{U}$
and concludes the proof. ∎

Recall that ultrafilters can also be characterised by the Stone-Čech
compactification. More precisely, we have the following (see, _e.g._ , [34,
Chapter 7.4]):

######  Lemma A.7.

Let $X$ be a set and $\beta X$ be the Stone-Čech compactification of $X$.

  1. (1)

Given $\omega\in\beta X$, the family $\\{A\subseteq
X:\omega\in\overline{A}\\}$ is an ultrafilter on $X$.

  2. (2)

Given an ultrafilter $\mathcal{U}$ on $X$, the intersection
$\bigcap_{A\in\mathcal{U}}\overline{A}$ consists of a single point.

The procedures above are inverse to each other, and hence $\beta X$ can be
characterised by ultrafilters on $X$. Moreover, points in $\partial_{\beta}X$
correspond to non-principal ultrafilters.

Thanks to Lemma A.7, we will also use ultrafilters and points in the Stone-
Čech compactification freely without further explanation throughout the paper.

For convenience, we also record that for $D\subseteq X$, its closure
$\overline{D}$ in $\beta X$ satisfies:

$\overline{D}=\\{\omega\in\beta X:\omega(D)=1\\}$

and $\overline{D}$ is clopen in $\beta X$. The topology of $\beta X$ is
generated by $\\{\overline{D}:D\subseteq X\\}$.

Finally we recall the following, which can be proved either directly using the
universal property of the Stone-Čech compactification or deduced directly from
Lemma A.4, A.5 and A.7:

######  Corollary A.8.

For a subset $Z\subseteq X$, the closure $\overline{Z}$ in $\beta X$ is
homeomorphic to $\beta Z$.

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