# Independence of linear spectral statistics and the point process at the edge
of Wigner matrices

Debapratim Banerjee
Indian Statistical Institute, Kolkata
<EMAIL_ADDRESS>

###### Abstract

In the current paper we consider a Wigner matrix and consider an analytic
function of polynomial growth on a set containing the support of the
semicircular law in its interior. We prove that the linear spectral statistics
corresponding to the function and the point process at the edge of the Wigner
matrix are asymptotically independent when the entries of the Wigner matrix
are sub-Gaussian. The main ingredient of the proof is based on a recent paper
by Banerjee [6]. The result of this paper can be viewed as a first step to
find the joint distribution of eigenvalues in the bulk and the edge.

##  1 Introduction

Since the groundbreaking discovery of Wigner [16], Wigner matrices have been a
topic of key interest in the mathematics and physics communities. Later on
these matrices proved to be important for many models in engineering, high
dimensional statistics and many other branches.

In particular, a Wigner matrix is a $n\times n$ symmetric (hermitian) matrix
with real (complex) entries where the entries of the upper diagonal part are
i.i.d. with mean $0$ and variance $\frac{1}{n}$. One is interested in the
eigenvalue distribution of the matrix when the dimension grows to infinity.
The study of eigenvalues of Wigner matrices started with characterizing the
limiting distribution of the histogram of the eigenvalues. This is done in the
seminal papers of Wigner [16] and Wigner [15]. It is known that this limiting
distribution exists and coined as the famous semicircular distribution. In
particular, it is given by the following density.

$f(x)=\left\\{\begin{array}[]{ll}\frac{1}{2\pi}\sqrt{4-x^{2}}&\text{when\leavevmode\nobreak\
$|x|\leq 2$}\\\ 0&\text{otherwise}\end{array}\right.$ (1.1)

However after specifying the spectral distribution, there has been a
remarkable advancement in this topic. Several interesting questions related to
the eigenvalues have been raised and have been subsequently solved.

The main objective of this paper is to relate two such branches. The first one
is the study of $\operatorname{Tr}[g(W)]$ for an analytic function $g$
containing the support of the semicircular law and the second one is the study
of eigenvalues near the support of the semicircular law. Here $W$ is the
Wigner matrix. The second one is about the study of the eigenvalues near the
edge of the spectrum.

The study of linear spectral statistics was initiated in the paper of Bai and
Silverstein [4]. Here one considers a generalized Wishart matrix and the
linear spectral statistics of a general analytic function. There have been
several papers in this topic after that. For Wigner matrices the CLT of linear
spectral statistics was studied in Bai et al. [3]. These results are proven
using a rigorous analysis of the Stieltjes transform of the empirical
distribution function of the eigenvalues of the Wigner/Wishart matrix. On the
other hand a combinatorial approach for solving the similar problem was
initiated in the paper by Anderson and Zeitouni [1].

On the other hand, study of the eigenvalues near the edge of the Wigner matrix
is also well known. Here we look at the point processes of the eigenvalues
near $\pm 2$ which is the support of the semicircular distribution. Using the
explicit distribution of the eigenvalues the fluctuation of the largest
eigenvalue of the Wigner matrix was first proved in the seminal works of Tracy
and Widom [12], [13] in the Gaussian case. In particular it is proved that

$\mathbb{P}\left[n^{\frac{2}{3}}\left(\lambda_{1,n}-2\right)\leq s\right]\to
F_{\beta}(s)$ (1.2)

where the Tracy-Widom distribution functions $F_{\beta}$ can be described by
Painleve equations, and $\beta=1,2,4$ corresponds to
Orthogonal/Unitary/Symplectic ensemble, respectively. Here
$\lambda_{1,n}\geq\ldots\geq\lambda_{n,n}$ are the eigenvalues of the Wigner
matrix $W$. The joint distribution of $k$ largest eigenvalues can be expressed
in terms of the Airy kernel, which was shown by Forrester [8]. In general the
joint distribution of $\left(\lambda_{1,n},\ldots,\lambda_{k,n}\right)$ will
also converge after proper rescaling and centering. In particular

$\begin{split}\mathbb{P}\left[n^{\frac{2}{3}}\left(\lambda_{1,n}-2\right)\leq
s_{1},\ldots,n^{\frac{2}{3}}\left(\lambda_{k,n}-2\right)\leq s_{k}\right]\to
F_{\beta,k}(s_{1},\ldots,s_{k}).\end{split}$ (1.3)

The $k$ dimensional distribution $F_{\beta,k}$ will also be coined as Tracy
Widom distribution.

After the Gaussian case being solved, a significant amount of work has been
done for general non-Gaussian entries. This problem is known as edge
universality. The edge universality of the Wigner matrix was first proved in
the paper Soshnikov [10] through combinatorial methods. He assumed that the
distributions of the entries of the matrix are sub-Gaussian and symmetric. The
universality for the non-symmetric entries was first proved by Tao and Vu
[11]. Here one assumes that entries have vanishing third moment and the tail
decays exponentially. Then there was a different approach by Erdős, Yau and
others by analyzing the resolvent matrix. One might look at Erdős and Yau [7].
Very recently Banerjee [6] proved edge universality through a new
combinatorial approach where he removed assumption of symmetry of the entries.

However to the best of our limited knowledge nothing has been proved for the
joint distribution of a generic linear spectral statistics and the eigenvalues
near the edge. _In this paper following the combinatorial approach of Banerjee
[6] and Anderson and Zeitouni [1] we prove that they are asymptotically
independent._ This result can be seen as a first step to prove the joint
distribution of eigenvalues near a typical point in the interior of the
support of the semicircular law and the eigenvalues near the edge. This is due
to the following reason: we choose a point $x\in(-2,2)$ and look at an
interval of length $O\left(\frac{1}{n}\right)$ around $x$ and look at the
eigenvalues falling in the interval. This type of statistics can be
approximated by a function $g_{x,n}(\cdot)$ where $g_{x,n}(\cdot)$ takes
$O(1)$ values in an interval of length $O\left(\frac{1}{n}\right)$ around $x$
and is very close to $0$ outside the interval. However as of now, there are
some technical difficulties in this scenario. This is due to the fact that the
main ingredient of the proof is by approximation of an analytic function by a
first few terms of it’s power series expansion. However for analytic functions
just discussed, this kind of approximation is not well behaved. This is an
ongoing work.

The linear spectral statistics and the largest eigenvalue are also interesting
from high dimensional statistics and spin glass point of view. In spin
glasses, it is known that for spherical Sherrington Kirkpatrick model the free
energy at high temperature can be approximated by a linear spectral statistics
of the interaction matrix and in low temperature it can be approximated by the
largest eigenvalue of the interaction matrix. One might look at Baik and Lee
[5] for a reference. A straight forward corollary of our result is the
independence of free energies in low and high temperature of spherical SK
model made out of same interaction matrix. The joint distribution of largest
eigenvalue and a generic linear spectral statistics is also important in high
dimensional statistics. In particular it is proved in Zhang et al. [17] for
large spikes, when we consider the largest eigenvalue of a spiked Wishart
matrix and a generic spectral statistics they are independent. One might also
look at Li et al. [9]. However these results are for the spiked matrices where
the spikes are large. In particular, the largest eigenvalue has an asymptotic
normal distribution as opposed to Tracy-Widom distribution.

##  2 Preliminaries

###  2.1 Model description and related terminologies

Firstly we start with the definition of Wigner matrices.

######  Definition 2.1.

We call a matrix $W=\left(x_{i,j}/\sqrt{n}\right)_{1\leq i,j\leq n}$ to be a
Wigner matrix if $x_{i,j}=\bar{x}_{j,i}$, $\left(x_{i,j}\right)_{1\leq i<j\leq
n}$ are i.i.d., $\operatorname{E}[x_{i,j}]=0$ and
$\operatorname{E}[|x_{i,j}|^{2}]=1$.

In this paper we deal with the real symmetric matrices and for the ease of
calculation we scale the whole matrix by a factor $2$. With slight abuse of
notation we shall also call this matrix a Wigner matrix and denote it by $W$.
Following are the assumptions of the matrices we consider.

######  Assumption 2.1.

We consider a matrix $W$ given by $W=(x_{i,j}/\sqrt{n})_{1\leq i,j\leq n}$
such that the following conditions are satisfied:

  1. (i)

$x_{i,j}=x_{j,i}$ for $i\leq j$.

  2. (ii)

$\operatorname{Var}(x_{i,j})=\frac{1}{4}$

  3. (iii)

$\left(x_{i,j}\right)_{1\leq i<j\leq n}$ are i.i.d.

  4. (iv)

$\operatorname{E}[x_{i,j}^{2k}]\leq\left(\mathrm{const}.k\right)^{k}$
$\forall\leavevmode\nobreak\ k\in\mathbb{N}$

Given a Wigner matrix $W$ of dimension $n\times n$ we denote its eigenvalues
by $\lambda_{1,n}\geq\ldots\geq\lambda_{n,n}$. It is well known that for a
Wigner matrix in Definition 2.1, the measure
$\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_{i,n}}$ converge weakly to the
semicircular law in the almost sure sense. The law is given by the following
density:

$f(x)=\left\\{\begin{array}[]{ll}\frac{1}{2\pi}\sqrt{4-x^{2}}&\text{when\leavevmode\nobreak\
$|x|\leq 2$}\\\ 0&\text{otherwise.}\end{array}\right.$ (2.1)

When we scale the entries by a factor $2$, the distribution is supported in
$[-1,1]$ and its density is given by

$f(x)=\left\\{\begin{array}[]{ll}\frac{2}{\pi}\sqrt{1-x^{2}}&\text{when\leavevmode\nobreak\
$|x|\leq 1$}\\\ 0&\text{otherwise.}\end{array}\right.$ (2.2)

One of the main quantities of consideration in this paper is the linear
spectral statistics. This is defined as follows:

######  Definition 2.2.

Suppose $g$ is an analytic function on a set containing $[-1,1]$ in its
interior. We define the linear spectral statistics corresponding to the
function $g$ as follows:

$\operatorname{Tr}[g(W)]=\sum_{i=1}^{n}g(\lambda_{i,n})$ (2.3)

Our main approach is to work with a polynomial function and use the power
series representation of the polynomial function to work with general analytic
function. So most of the time $g$ will be a polynomial of finite degree.

###  2.2 Preliminaries for the combinatorial approach

The main approach of this paper is combinatorial. For the linear spectral
statistics we follow the approach of Anderson and Zeitouni [1] and for the
point process near the edge we follow the approach of Banerjee [6]. In this
subsection we develop the required terminologies. To begin with we start with
a matrix $W$ of dimension $n\times n$. Its $k$ th moment is given by

$\operatorname{Tr}[W^{k}]=\sum_{i_{0},i_{1},\ldots,i_{k-1},i_{0}}W_{i_{0},i_{1}}\ldots
W_{i_{k-1},i_{0}}.$ (2.4)

The word sentence method systematically analyzes the tuples
$(i_{0},\ldots,i_{k-1},i_{0})$ for some suitable $k$. To do this we need some
notations and definitions. In this part we give a very brief introduction to
words, sentences and their equivalence classes essential for the combinatorial
analysis of random matrices. The definitions are taken from Anderson et al.
[2] and Anderson and Zeitouni [1]. For more general information, see [2,
Chapter 1] and [1].

######  Definition 2.3 ($\mathcal{S}$ words).

Given a set $\mathcal{S}$, an $\mathcal{S}$ letter $s$ is simply an element of
$\mathcal{S}$. An $\mathcal{S}$ word $w$ is a finite sequence of letters
$s_{1}\ldots s_{k}$, at least one letter long. An $\mathcal{S}$ word $w$ is
_closed_ if its first and last letters are the same. In this paper
$\mathcal{S}=\\{1,\ldots,n\\}$ where $n$ is the number of nodes in the graph.

Two $\mathcal{S}$ words $w_{1},w_{2}$ are called _equivalent_ , denoted
$w_{1}\sim w_{2}$, if there is a bijection on $\mathcal{S}$ that maps one into
the other. For any word $w=s_{1}\ldots s_{k}$, we use $l(w)=k$ to denote the
_length_ of $w$, define the _weight_ $wt(w)$ as the number of distinct
elements of the set ${s_{1},\ldots,s_{k}}$ and the _support_ of $w$, denoted
by $\mathrm{supp}(w)$, as the set of letters appearing in $w$. With any word
$w$ we may associate an undirected graph, with $wt(w)$ vertices and at most
$l(w)-1$ edges, as follows.

######  Definition 2.4 (Graph associated with a word).

Given a word $w=s_{1}\ldots s_{k}$, we let $G_{w}=(V_{w},E_{w})$ be the graph
with set of vertices $V_{w}=\mathrm{supp}(w)$ and (undirected) edges
$E_{w}=\\{\\{s_{i},s_{i+1}\\},i=1,\ldots,k-1\\}.$

The graph $G_{w}$ is connected since the word $w$ defines a path connecting
all the vertices of $G_{w}$, which further starts and terminates at the same
vertex if the word is closed. We note that equivalent words generate the same
graphs $G_{w}$ (up to graph isomorphism) and the same passage-counts of the
edges. Given an equivalence class $\mathbf{w}$, we shall sometimes denote
$\\#E_{\mathbf{w}}$ and $\\#V_{\mathbf{w}}$ to be the common number of edges
and vertices for graphs associated with all the words in this equivalence
class $\mathbf{w}$.

######  Definition 2.5 (Weak Wigner words).

Any word $w$ will be called a _weak Wigner word_ if the following conditions
are satisfied:

  1. 1.

$w$ is closed.

  2. 2.

$w$ visits every edge in $G_{w}$ at least twice.

Suppose now that $w$ is a weak Wigner word. If $wt(w)=(l(w)+1)/2$, then we
drop the modifier “weak” and call $w$ a _Wigner word_. (Every single letter
word is automatically a Wigner word.) Except for single letter words, each
edge in a Wigner word is traversed exactly twice. If $wt(w)=(l(w)-1)/2$, then
we call $w$ a _critical weak Wigner word_.

It is a well known result in random matrix theory that there is a bijection
from the set of the Wigner words of length $2k+1$ to the set of Dyck paths of
length $2k$. We now move to definitions related to sentences.

######  Definition 2.6 (Sentences and corresponding graphs).

A sentence $a=[w_{i}]_{i=1}^{m}=[[\alpha_{i,j}]_{j=1}^{l(w_{i})}]_{i=1}^{m}$
is an ordered collection of $m$ words of length $(l(w_{1}),\ldots,l(w_{m}))$
respectively. We define the graph $G_{a}=(V_{a},E_{a})$ to be the graph with

$V_{a}=\mathrm{supp}(a),\quad
E_{a}=\left\\{\\{\alpha_{i,j},\alpha_{i,j+1}\\}|i=1,\ldots,m;j=1,\ldots,l(w_{i})-1\\}\right\\}.$

######  Notation 2.1.

Given a closed word $w=(i_{0},i_{1},\ldots,i_{k-1},i_{k}=i_{0})$ we need to
consider the random variable $\prod_{j=0}^{k-1}x_{i_{j},i_{j+1}}$. We call
this random variable $X_{w}$.

In addition to this we also need the map from word to Dyck paths and the
related results and terminologies for the analysis of the eigenvalues near the
edge. For these the reader may have a look at subsection 7.2 in Banerjee [6].

##  3 Main result

We now state the main result of this paper.

######  Theorem 3.1.

Suppose we have a Wigner matrix $W$ satisfying the assumptions in Assumption
2.1. Then the following are true:

  1. 1.

Let $g$ be any polynomial of fixed degree. Then the random variable
$\left(\operatorname{Tr}[g(W)]-\operatorname{E}\left[\operatorname{Tr}[g(w)]\right]\right)$
and the point process at the edge of the spectrum (i.e. at $x=\pm 1$) are
asymptotically independent.

  2. 2.

More generally $g$ can be taken to be a fixed analytic function of polynomial
growth on a set containing $[-1,1]$ in its interior.

  3. 3.

As a corollary, for any fixed $k$, the random vector
$\left(n^{\frac{2}{3}}2\left(\lambda_{1,n}-1\right),\ldots,n^{\frac{2}{3}}2\left(\lambda_{k,n}-1\right)\right)$
and
$\left(\operatorname{Tr}[g(W)]-\operatorname{E}\left[\operatorname{Tr}[g(w)]\right]\right)$
are independent.

##  4 Strategies for the proof

As mentioned at the beginning, the main technique of this current paper is
combinatorial in nature. In particular for an analytic function $g$, it is
known that
$\left(\operatorname{Tr}\left[g(W)\right]-\operatorname{E}\left[\operatorname{Tr}\left[g(W)\right]\right]\right)$
converges to a centered Gaussian with explicit variance. The combinatorial
approach for proving this kind of result was initiated in the paper by
Anderson and Zeitouni [1]. In order the prove this the authors first consider
the joint distribution of

$\left(\operatorname{Tr}\left[W^{m_{1}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{1}}\right]\right],\ldots,\operatorname{Tr}\left[W^{m_{k}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{k}}\right]\right]\right)$

for every fixed $k$ and integers $m_{1},\ldots,m_{k}\in\mathbb{N}$. It is
proved using the method of moments that this joint distribution is $k$
dimensional Gaussian with explicit covariance matrix. The results are
summarized in the following theorem.

######  Theorem 4.1.

(Chapter $1$ Anderson et al. [2]) Suppose $W$ is a Wigner matrix satisfying
Assumption 2.1 and we want to determine the joint distribution of

$\left(\operatorname{Tr}\left[W^{m_{1}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{1}}\right]\right],\ldots,\operatorname{Tr}\left[W^{m_{k}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{k}}\right]\right]\right).$

Then there exists $\Sigma$ such that

  1. 1.

for any $l$ and
$m_{1}^{\prime},\ldots,m_{l}^{\prime}\in\\{m_{1},\ldots,m_{k}\\}$ the joint
moment

$\begin{split}&\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\right]\\\
&\to\left\\{\begin{array}[]{ll}0&\text{if $l$ is odd}\\\
\sum_{\eta\in\mathcal{P}_{l}}\prod_{j=1}^{\frac{l}{2}}\Sigma\left(m_{\eta(j,1)}^{\prime},m_{\eta(j,2)}^{\prime}\right)&\text{otherwise}.\end{array}\right.\end{split}$
(4.1)

Here for an even number $l$, $\mathcal{P}_{l}$ denotes all partitions of the
set $\\{1,2,\ldots,l\\}$ such that each block has exactly two elements.
Further for any partition $\eta\in\mathcal{P}_{l}$, $\eta(j,1)$ and
$\eta(j,2)$ denote the first and second element of the $j$ th block in $\eta$.

  2. 2.

As a corollary, we have the vector

$\left(\operatorname{Tr}\left[W^{m_{1}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{1}}\right]\right],\ldots,\operatorname{Tr}\left[W^{m_{k}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{k}}\right]\right]\right)$

converges to a $k$ dimensional centered Gaussian with covariance matrix
$\left(\Sigma(m_{i},m_{j})\right)_{1\leq i,j\leq k}$.

For an explicit expression of $\Sigma$ one might look at section 2.1.7 of
Anderson et al. [2].

Finally for a general analytic function $g(x)=\sum_{i=0}^{\infty}g_{i}x^{i}$,
one considers the following polynomial $g^{(m)}(x)=\sum_{i=0}^{m}g_{i}x^{i}$
and prove the following result:

######  Theorem 4.2.

(Anderson and Zeitouni [1]) Suppose $W$ be a Wigner matrix satisfying
Assumption 2.1. Then for any analytic function
$g(x)=\sum_{i=0}^{\infty}g_{i}x^{i}$ on a set containing
$[-1-\varepsilon,1+\varepsilon]$ and having polynomial growth we have

$\operatorname{Var}\left[\operatorname{Tr}[g(W)]\right]\leq\bar{c}\sup_{x\in[-1-\varepsilon,1+\varepsilon]}|g^{\prime}(x)|^{2}.$
(4.2)

In particular if the function is analytic on a interval containing
$[-1-\varepsilon,1+\varepsilon]$, for any $\eta$ we can choose large enough
$i$ such that

$\operatorname{Var}\left[\operatorname{Tr}[g(W)]-\sum_{j=0}^{i}g_{j}x^{j}\right]\leq\eta.$
(4.3)

######  Remark 4.1.

One might note that for Theorem 4.2 to hold, Anderson and Zeitouni [1] require
the entries to satisfy Poincare inequality. However one might look at Remark
3.12 of Van Handel [14] to see that sub-Gaussianity implies Poincare
inequality.

On the other hand in the recent paper Banerjee [6] the combinatorial approach
for proving edge universality with general non-symmetrically distributed
entries was considered. This method relies upon proving the following results

######  Theorem 4.3.

(Banerjee [6]) Consider the Wigner matrix $W$ satisfying Assumption 2.1. Then
for any fixed $t\in(0,\infty)$ taking $k=\left[tn^{\frac{2}{3}}\right]$, we
have the following results

  1. 1.

$\operatorname{E}\operatorname{Tr}\left[W^{2k}\right]=O(1)$ and
$\operatorname{E}\operatorname{Tr}\left[W^{2k+1}\right]=o(1)$.

  2. 2.

If the limit of
$\lim_{n\to\infty}\operatorname{E}\operatorname{Tr}\left[W^{2k}\right]$ for
some $t\in(0,\infty)$ exists, then the limit only depends on the first and
second moment of entries.

  3. 3.

As the limit exists for Gaussian entries, the limit exists and is universal
for any Wigner matrix satisfying Assumption 2.1.

######  Theorem 4.4.

(Banerjee [6]) Consider the Wigner matrix $W$ satisfying Assumption 2.1. Then
for any fixed $t_{1},\ldots,t_{l}\in(0,\infty)^{l}$ taking
$k_{i}=\left[t_{i}n^{\frac{2}{3}}\right]$, we have the following results

  1. 1.

$\operatorname{E}\left[\prod_{i=1}^{l}\left[\operatorname{Tr}\left[W^{k_{i}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{k_{i}}\right]\right]\right]\right]=O(1).$
(4.4)

  2. 2.

If the limit in (4.4) exists for some $t_{1},\ldots,t_{l}$, then the limit
only depends on the first and second moment of entries.

  3. 3.

As the limit exists for Gaussian entries, the limit exists and is universal
for any Wigner matrix satisfying Assumption 2.1.

Theorems 4.3 and 4.4 is enough to prove the edge universality as the laplace
transformation of the $l$ point correlation function of the point process at
the edge of the spectrum is determined by the limits in Theorems 4.4 and 4.3.
As weak convergence of correlation functions of a sequence of point processes
determines the weak convergence of the sequence of point processes, we get the
desired result of edge universality.

So in order to prove the independence of the linear spectral statistics and
the point process at the edge, we prove the following result:

######  Theorem 4.5.

Suppose we fix $m_{1},\ldots,m_{k}$. Then for any $l$, $l^{\prime}$,
$m_{1}^{\prime},\ldots,m_{l}^{\prime}\in\\{m_{1},\ldots,m_{k}\\}$ and
$t_{1},\ldots,t_{l^{\prime}}\in(0,\infty)$, the following holds:

$\begin{split}&\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\\\
&\to\lim_{n\to\infty}\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\right]\lim_{n\to\infty}\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\end{split}$
(4.5)

Once this is proved, Theorem 3.1 readily follows with the help of Theorem 4.2.

##  5 Proofs of the results

To begin with we state an algorithm. This algorithm takes two closed words
$w_{1}$ and $w_{2}$ of lengths $k_{1}+1$ and $k_{2}+1$ as input such that the
words have at least one edge in common and gives a closed word $w_{3}$ of
length $k_{1}+k_{2}+1$ as an output which has the same edge set as the union
of the edges of $w_{1}$ and $w_{2}$. This is taken from Banerjee [6]

######  Algorithm 5.1.

We start with two words $w_{1}$ and $w_{2}$ such that $w_{1}$ and $w_{2}$
shares an edge. Let $\\{\alpha,\beta\\}$ be the first edge in $w_{2}$ which is
repeated in $w_{1}$. We consider the first appearance of $\\{\alpha,\beta\\}$
in $w_{2}$. Without loss of generality we assume that the first appearance of
the edge $\\{\alpha,\beta\\}$ appears in the word $w_{2}$ in the order
$(\alpha,\beta)$. We now consider any appearance (for concreteness say the
first) of the edge $\\{\alpha,\beta\\}$ in the word $w_{1}$. This appearance
$\\{\alpha,\beta\\}$ can be traversed in $w_{1}$ in the order $(\alpha,\beta)$
or $(\beta,\alpha)$. Considering these we have the word $w_{2}$ looks like

$w_{2}=(\alpha_{0},\alpha_{1},\ldots,\alpha_{p_{1}},\alpha,\beta,\ldots,\alpha_{k_{1}-1},\alpha_{0})$
(5.1)

and the word $w_{1}$ looks like

$w_{1}=\left(\beta_{0},\beta_{1},\ldots,\beta_{q_{1}},\alpha,\beta,\ldots,\beta_{k_{2}-1},\beta_{0}\right)$
(5.2)

or

$w_{1}=(\beta_{0},\beta_{1},\ldots,\beta_{q_{1}},\beta,\alpha,\ldots,\beta_{k_{2}-1},\beta_{0}).$
(5.3)

Now we output the word $w_{3}$ as follows:

  1. 1.

Suppose $w_{1}$ is of the form (5.2), then

$w_{3}=\left(\alpha_{0},\alpha_{1},\ldots,\alpha_{p_{1}},\alpha,\beta,\beta_{q_{1}+3},\ldots,\beta_{k_{2}-1},\beta_{0},\ldots,\beta_{q_{1}},\alpha,\beta,\alpha_{p_{1}+3},\ldots,\alpha_{k_{1}-1},\alpha_{0}\right).$
(5.4)

  2. 2.

On the other hand when $w_{1}$ is of the form (5.3),

$w_{3}=\left(\alpha_{0},\alpha_{1},\ldots,\alpha_{p_{1}},\alpha,\beta,\beta_{q_{1}},\ldots,\beta_{0},\beta_{k_{2}-1},\ldots,\beta_{q_{1}+3},\alpha,\beta,\alpha_{p_{1}+3},\ldots,\alpha_{k_{1}-1},\alpha_{0}\right).$
(5.5)

###### Proof of Theorem 4.5.

As this is the most important proof in this paper, we prove it in step by
step.
Step 1 (A basic observation): We begin with a very basic observation.

$\begin{split}&\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\\\
&=\left(\frac{1}{n}\right)^{\sum_{i}\frac{m_{i}^{\prime}}{2}+\sum_{j}\frac{[t_{j}n^{\frac{2}{3}}]}{2}}\sum_{w_{1},\ldots,w_{l}\leavevmode\nobreak\
|\leavevmode\nobreak\
l(w_{i})=m_{i}^{\prime}+1}\sum_{w_{1}^{\prime},\ldots,w_{l^{\prime}}^{\prime}\leavevmode\nobreak\
|\leavevmode\nobreak\
l(w_{j}^{\prime})=[t_{j}n^{\frac{2}{3}}]+1}\operatorname{E}\left[\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)\right]\\\
&=\left(\frac{1}{n}\right)^{\sum_{i}\frac{m_{i}^{\prime}}{2}+\sum_{j}\frac{[t_{j}n^{\frac{2}{3}}]}{2}}\sum_{a_{1}}\sum_{a_{2}}\operatorname{E}\left[\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)\right]\\\
&=\underbrace{\left(\frac{1}{n}\right)^{\sum_{i}\frac{m_{i}^{\prime}}{2}+\sum_{j}\frac{[t_{j}n^{\frac{2}{3}}]}{2}}\left(\sum_{a_{1},a_{2}\leavevmode\nobreak\
|\leavevmode\nobreak\ E_{a_{1}}\cap
E_{a_{2}\neq\emptyset}}\operatorname{E}\left[\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)\right]\right)}_{T_{1}(say)}+\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\underbrace{\left(\frac{1}{n}\right)^{\sum_{i}\frac{m_{i}^{\prime}}{2}+\sum_{j}\frac{[t_{j}n^{\frac{2}{3}}]}{2}}\left(\sum_{a_{1},a_{2}\leavevmode\nobreak\
|\leavevmode\nobreak\ E_{a_{1}}\cap
E_{a_{2}=\emptyset}}\operatorname{E}\left[\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)\right]\right)}_{T_{2}(say)}\end{split}$
(5.6)

Here $a_{1}=[w_{i}]_{i=1}^{l}$ and
$a_{2}=[w_{j}^{\prime}]_{j=1}^{l^{\prime}}$. We shall analyze $T_{1}$ and
$T_{2}$ separately.
Step 2 (Analysis of $T_{1}$) $T_{1}$ contains the cases when $E_{a_{1}}\cap
E_{a_{2}}\neq\emptyset$.

For this without loss of generality we assume $a_{1}$ contains a single word
and so does $a_{2}$. The general case can be proved by repeated use of the
arguments given next. We assume $w_{1}$ be a word of fixed length $m+1$ while
$w_{2}$ is a word of length $[tn^{\frac{2}{3}}]+1$ for some $t\in(0,\infty)$
such that $E_{w_{1}}\cap E_{w_{2}}\neq\emptyset$. We would like to find the
following expectation

$\begin{split}\left(\frac{1}{n}\right)^{\frac{m+[tn^{\frac{2}{3}}]}{2}}\sum_{w_{1},w_{2}\leavevmode\nobreak\
|\leavevmode\nobreak\ l(w_{1})=m+1,l(w_{2})=[tn^{\frac{2}{3}}]+1,E_{w_{1}}\cap
E_{w_{2}}\neq\emptyset}\left(\operatorname{E}[X_{w_{1}}X_{w_{2}}]-\operatorname{E}[X_{w_{1}}]\operatorname{E}[X_{w_{2}}]\right)\end{split}$
(5.7)

We now apply Algorithm 5.1 to $(w_{1},w_{2})$ to get a word $\mathfrak{w}$.
Hence we write the sum in (5.7) in the following way:

$\left(\frac{1}{n}\right)^{\frac{m+[tn^{\frac{2}{3}}]}{2}}\sum_{\mathfrak{w}}\sum_{w_{1},w_{2}\leavevmode\nobreak\
|\leavevmode\nobreak\
f(w_{1},w_{2})=\mathfrak{w}}\left(\operatorname{E}[X_{w_{1}}X_{w_{2}}]-\operatorname{E}[X_{w_{1}}]\operatorname{E}[X_{w_{2}}]\right).$
(5.8)

In order to have a nontrivial value of
$\left(\operatorname{E}[X_{w_{1}}X_{w_{2}}]-\operatorname{E}[X_{w_{1}}]\operatorname{E}[X_{w_{2}}]\right)$,
one needs to have each edge in $\mathfrak{w}$ repeated at least twice. We call
this set $\mathcal{W}_{\geq 2,m+[tn^{\frac{2}{3}}]}$. By a straight forward
application of Holder’s inequality we know
$\operatorname{E}[|X_{w_{1}}X_{w_{2}}|]\geq\operatorname{E}[|X_{w_{1}}|]\operatorname{E}[|X_{w_{2}}|].$
So it is enough to bound the following quantity:

$\left(\frac{1}{n}\right)^{\frac{m+[tn^{\frac{2}{3}}]}{2}}\sum_{\mathfrak{w}\in\mathcal{W}_{\geq
2,m+[tn^{\frac{2}{3}}]}}2|f^{-1}(\mathfrak{w})|\operatorname{E}[|X_{\mathfrak{w}}|].$
(5.9)

We now look closely at the proof of Theorem 6.2 in Banerjee [6]. Imitating the
arguments of the proof of this theorem it can be showed that all the other
cases apart from case (i) have negligible contribution. Also following the
proofs of Propositions 8.2 and 8.3 in Banerjee [6], we can restrict our
attention to the case when every edge in $\mathfrak{w}$ is traversed exactly
twice and every type $j\geq 2$ instant in the skeleton word is actually a type
$2$ instant. Now we analyze this case in details.

Alike Banerjee [6] we at first fix the skeleton word. Let $N$ be the number of
type $j\geq 2$ instants in the skeleton word. Given a skeleton word, we choose
a type $j\geq 2$ instant and find out the edge which is closed in the next
instant. We choose the instant which corresponds to the left endpoint of the
first traversal of this edge. So given the skeleton word there is at most $N$
choices to choose the endpoints of $w_{1}$. Let this choice be $l$. Let the
length of the part of the skeleton word inside these two points be $m_{1}$ and
the outside these two points be $m_{2}$. As a consequence, in $w_{2}$ there
will be exactly $m_{2}+1$ Dyck paths and in $w_{1}$ there will be exactly
$m_{1}$ Dyck paths. So this might appear same as the calculation in Theorem
6.2 in Banerjee [6]. However as the length of the word $w_{1}$ is fixed, there
will be additional constraints. In particular, if the first endpoint of the
word is the instant where this level first appears in the skeleton word, we
have $|p_{1}+\ldots+p_{l}-r_{l}|\leq m$. Here we chose the $l$ th type $j\geq
2$ instant. On the other hand, it is possible that the skeleton word returns
to the level of first chosen point multiple times before reaching the second
chosen point. In this case we have $p_{l}\leq m$. As all the type $j\geq 2$
instants in the skeleton word are type $2$, we have the following upper bound
to (5.9):

$\begin{split}&C\left(\frac{1}{2}\right)^{m+[tn^{\frac{2}{3}}]}\sum_{N}\sum_{l=1}^{N}\sum_{p_{1},\ldots,p_{N}}\sum_{q_{1},\ldots,q_{N-1}}\sum_{r_{1},\ldots,r_{N}\leavevmode\nobreak\
:\leavevmode\nobreak\ |p_{1}+\ldots+p_{l}-r_{l}|\leq
m}n^{\frac{m+[tn^{\frac{2}{3}}]}{2}-N+1}3^{N}\\\ &\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\left(m+1\right)\frac{m_{1}}{m+1}\binom{m+1}{\frac{m_{1}+m+1}{2}}\frac{m_{2}+1}{[tn^{\frac{2}{3}}]+1}\binom{[tn^{\frac{2}{3}}]+1}{\frac{m_{2}+[tn^{\frac{2}{3}}]+2}{2}}\\\
&+C\left(\frac{1}{2}\right)^{m+[tn^{\frac{2}{3}}]}\sum_{N}\sum_{l=1}^{N}\sum_{p_{1},\ldots,p_{N}\leavevmode\nobreak\
|\leavevmode\nobreak\ p_{l}\leq
m}\sum_{q_{1},\ldots,q_{N-1}}\sum_{r_{1},\ldots,r_{N}}n^{\frac{m+[tn^{\frac{2}{3}}]}{2}-N+1}3^{N}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left(m+1\right)\frac{m_{1}}{m+1}\binom{m+1}{\frac{m_{1}+m+1}{2}}\frac{m_{2}+1}{[tn^{\frac{2}{3}}]+1}\binom{[tn^{\frac{2}{3}}]+1}{\frac{m_{2}+[tn^{\frac{2}{3}}]+2}{2}}+o(1)\end{split}$
(5.10)

As the sum over all the unconstrained case (i.e. when both $l(w_{1})$ and
$l(w_{2})$ is of $O(n^{\frac{2}{3}})$) is of $O(1)$, the sum in (5.10) goes to
$0$.
Step 3 (Analysis of $T_{2}$) Now we consider the case when $E_{a_{1}}\cap
E_{a_{2}}=\emptyset$. Observe that in this case the random variables
$\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)$ and
$\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)$
are independent. As a result,

$\begin{split}&\operatorname{E}\left[\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)\right]\\\
&=\operatorname{E}\left[\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)\right]\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)\right]\end{split}$
(5.11)

By analysis of $T_{1}$,

$\begin{split}&\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\\\
&=\left(\frac{1}{n}\right)^{\sum_{i}\frac{m_{i}^{\prime}}{2}+\sum_{j}\frac{[t_{j}n^{\frac{2}{3}}]}{2}}\sum_{a_{1},a_{2}\leavevmode\nobreak\
|\leavevmode\nobreak\ E_{a_{1}}\cap
E_{a_{2}=\emptyset}}\operatorname{E}\left[\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)\right]\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)\right]+o(1).\end{split}$
(5.12)

Now we know that

$\begin{split}&\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\right]\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\\\
&=\left(\frac{1}{n}\right)^{\sum_{i}\frac{m_{i}^{\prime}}{2}+\sum_{j}\frac{[t_{j}n^{\frac{2}{3}}]}{2}}\sum_{a_{1},a_{2}}\operatorname{E}\left[\prod_{i=1}^{l}\left(X_{w_{i}}-E[X_{w_{i}}]\right)\right]\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(X_{w^{\prime}_{j}}-\operatorname{E}[X_{w^{\prime}_{j}}]\right)\right]\\\
&=T_{2}+T_{1}^{\prime}\end{split}$ (5.13)

where

$T_{1}^{\prime}=\sum_{a=[a_{1},a_{2}]\in\mathcal{A}_{\geq
2,s}}\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\right]\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right].$
(5.14)

Here $\mathcal{A}_{\geq 2,s}$ denotes all sentences $a=(a_{1},a_{2})$ such
that each edge in the graph $G_{a}$ is traversed at least twice and
$E_{a_{1}}\cap E_{a_{2}}\neq\emptyset.$ It is easy to see (by Holder’s
inequality) that $|T_{1}^{\prime}|\leq|T_{1}|$. Hence $|T_{1}^{\prime}|\to 0.$
As a consequence,

$\begin{split}&\left|\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]-\right.\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\left.\operatorname{E}\left[\prod_{i=1}^{l}\left(\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{m_{i}^{\prime}}\right]\right]\right)\right]\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\right|\to
0.\end{split}$ (5.15)

This completes the proof. ∎

###### Proof of Theorem 3.1.

We only prove part $2$ as proof of part $1$ is contained in the proof of
Theorem 4.5 and part $3$ is a straight forward corollary of part $2$.

This is enough to prove that for an analytic function $g$ of polynomial growth
on a set containing $[-1-\varepsilon,1+\varepsilon]$ and any bounded
continuous function $q:\mathbb{R}\to\mathbb{R}$ we have

$\begin{split}&\lim_{n\to\infty}\operatorname{E}\left[q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\\\
&=\lim_{n\to\infty}\operatorname{E}\left[q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)\right]\lim_{n\to\infty}\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\end{split}$
(5.16)

We shall fix any $\eta>0$ , choose $i$ large enough and compare
$q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)$
and
$q\left(\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right)$.
Here $g^{(i)}(x)=\sum_{j=0}^{i}g_{j}x^{j}$. Since both
$\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)$
is a Gaussian random variable in the limit and
$\operatorname{E}\left(\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right)^{2}$
is uniformly bounded over $n$ and $i$, they are tight uniformly over $n$ and
$i$. In particular we can choose $M$ large enough so that both
$P\left[\left|\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right|>M\right]$
and
$P\left[\left|\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right|>M\right]$
are less than $\eta$ uniformly over $n$ and $i$. Now inside $[-M,M]$ $q$ is
uniformly continuous. So given any $0<\eta^{\prime}<\eta$ there exists
$\delta>0$ such that $|x-y|<\delta$ implies $|q(x)-q(y)|<\eta^{\prime}$. Now
we choose $i$ large enough so that
$\operatorname{Var}\left[\operatorname{Tr}[g(W)]-\operatorname{Tr}[g^{(i)}(W)]\right]\leq\delta^{2}\eta$.
As a consequence, by Chebyshev’s inequality

$P\left[\left|q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)-q\left(\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right)\right|>\eta^{\prime}\right]<\eta+2\eta=3\eta.$

Hence,

$\operatorname{E}\left[\left(q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)-q\left(\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right)\right)^{2}\right]\leq
3\eta||q||_{\infty}^{2}+\eta^{2}$

. Now

$\begin{split}&\limsup_{n}\left|\operatorname{E}\left[q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]-\right.\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\left.\operatorname{E}\left[q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)\right]\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\right|\\\
&=\limsup_{n}\left|\operatorname{E}\left[q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]-\right.\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{E}\left[q\left(\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]+\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{E}\left[q\left(\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right)\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]-\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\left.\operatorname{E}\left[q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)\right]\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\right|\\\
&\leq\limsup_{n}\left|\operatorname{E}\left[q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)-q\left(\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right)\right]\times\right.\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\left.\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right|+\left|\operatorname{E}\left[\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right]\times\right.\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\left.\operatorname{E}\left[q\left(\operatorname{Tr}[g(W)]-\operatorname{E}[\operatorname{Tr}[g(W)]]\right)-q\left(\operatorname{Tr}[g^{(i)}(W)]-\operatorname{E}[\operatorname{Tr}[g^{(i)}(W)]]\right)\right]\right|+o(1)\\\
&\leq
2\sqrt{3\eta||q||_{\infty}^{2}+\eta^{2}}\limsup\left[\operatorname{E}\left(\prod_{j=1}^{l^{\prime}}\left(\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]-\operatorname{E}\left[\operatorname{Tr}\left[W^{[t_{j}n^{\frac{2}{3}}]}\right]\right]\right)\right)^{2}\right]^{\frac{1}{2}}+o(1)\end{split}$
(5.17)

Here the last line follows from Cauchy-Schwarz inequality. As $\eta>0$ was
arbitrary, this proves that the first expression of (5.17) goes to $0$. This
proves the result. ∎

Acknowledgment: This research is supported by an Inspire faculty fellowship. I
am highly grateful to Prof. Krishna Maddaly for his interest and
encouragement.

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# Interstellar Detection of 2-Cyanocyclopentadiene, C5H5CN, a Second Five-
Membered Ring Toward TMC-1

Kin Long Kelvin Lee  Department of Chemistry, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  P. Bryan Changala  Center for
Astrophysics $|$ Harvard & Smithsonian, Cambridge, MA 02138, USA  Ryan A.
Loomis  National Radio Astronomy Observatory, Charlottesville, VA 22903, USA
Andrew M. Burkhardt  Center for Astrophysics $|$ Harvard & Smithsonian,
Cambridge, MA 02138, USA  Ci Xue  Department of Chemistry, University of
Virginia, Charlottesville, VA 22904, USA  Martin A. Cordiner  Astrochemistry
Laboratory and the Goddard Center for Astrobiology, NASA Goddard Space Flight
Center, Greenbelt, MD 20771, USA  Institute for Astrophysics and Computational
Sciences, The Catholic University of America, Washington, DC 20064, USA
Steven B. Charnley  Astrochemistry Laboratory and the Goddard Center for
Astrobiology, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Michael C. McCarthy  Center for Astrophysics $|$ Harvard & Smithsonian,
Cambridge, MA 02138, USA  Brett A. McGuire  Department of Chemistry,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  National
Radio Astronomy Observatory, Charlottesville, VA 22903, USA  Center for
Astrophysics $|$ Harvard & Smithsonian, Cambridge, MA 02138, USA Kin Long
Kelvin Lee, Brett A. McGuire<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using radio observations with the Green Bank Telescope, evidence has now been
found for a second five-membered ring in the dense cloud Taurus Molecular
Cloud-1 (TMC-1). Based on additional observations of an ongoing, large-scale,
high-sensitivity spectral line survey (GOTHAM) at centimeter wavelengths
toward this source, we have used a combination of spectral stacking, Markov
chain Monte Carlo (MCMC), and matched filtering techniques to detect
2-cyanocyclopentadiene, a low-lying isomer of 1-cyanocyclopentadiene, which
was recently discovered there by the same methods. The new observational data
also yields a considerably improved detection significance for the more stable
isomer and evidence for several individual transitions between 23–32 GHz.
Through our MCMC analysis, we derive cospatial, total column densities of
$8.3\times 10^{11}$ and $1.9\times 10^{11}$ cm-2 for 1- and
2-cyanocyclopentadiene respectively, corresponding to a ratio of ${\sim}$4.4
favoring the former. The derived abundance ratios point towards a common
formation pathway—most likely being cyanation of cyclopentadiene by analogy to
benzonitrile.

Astrochemistry, ISM: molecules

##  1 Introduction

The recent astronomical detection of 1-cyano-1,3-cyclopentadiene (Fig. 1;
hereafter 1-cyano-CPD) and other CN-functionalized ring molecules in the
starless cloud core, Taurus Molecular Cloud-1 (TMC-1), has opened up an
entirely new and unexplored area of aromatic organic chemistry in space. In
rapid succession, evidence has been found for both five-membered (McCarthy et
al., 2020b) and six-membered rings (McGuire et al., 2018), and even bicyclic
ones (McGuire et al., 2020a) in a primordial gas cloud which has long been
known to be exceedingly rich in highly unsaturated carbon chains, most notably
cyanopolyynes and acetylenic free radicals, among others. Intriguingly, the
derived abundances of these rings exceed—in some cases by many orders of
magnitude—those predicted from chemical models which well reproduce the
abundance of a wide assortment of chains regardless of length. For this
reason, questions as to the relative importance of bottom-up formation
pathways versus inheritance from previous top-down routes that might survive
to the dense cloud phase have been raised, but remain poorly constrained at
present.

Substitution of a H atom of cyclopentadiene, $c$-C5H6 with a nitrile group
yields three possible cyanocyclopentadienes. Quantum chemical calculations by
McCarthy et al. (2020b), shown in Figure 1, predict 1-cyano-CPD is the most
stable isomer, followed closely by 2-cyano-1,3-cyclopentadiene (2-cyano-CPD; 5
kJ/mol or 600 K) and then 5-cyano-1,3-cyclopentadiene, which lies far higher
in energy (by 26 kJ/mol or ${\sim}$3130 K). These findings are in agreement
with those from an earlier study (Wentrup & Crow, 1970) and predicted from
conjugation arguments.

Figure 1: Geometric structures of the three low-lying cyanocyclopentadiene
isomers, 1-, 2-, and 5-cyanocyclopentadiene, $c$-C5H5CN and G3//B3LYP (Baboul
et al., 1999) energetics at 0 K. The dipole moment projections are $\mu_{a}$ =
4.15 D and $\mu_{b}$ = 0.27 D for 1-cyano-CPD, and $\mu_{a}$ = 4.36 D and
$\mu_{b}$ = 0.77 D for 2-cyano-CPD (Sakaizumi et al., 1987). 5-cyano-CPD has
not yet been observed, but possesses similarly favorable dipole moments
($\mu_{a}$ = 3.91 D, $\mu_{b}$ = 1.47 D calculated at the $\omega$B97X-D/cc-
pVQZ level of theory).

The rotational spectra of both 1- and 2-cyano-CPD have been reported by
several groups (Ford & Seitzman, 1978; Sakaizumi et al., 1987), with the most
recent set of measurements (McCarthy et al., 2020a, b) yielding very sharp
lines ($\sim$5 kHz FWHM linewidths) and rest frequencies accurate to 2 kHz or
better between 7 and 30 GHz. In terms of equivalent radial velocity, this
accuracy corresponds to $<0.1$ km s-1, which is more than sufficient to
conduct a rigorous search in the coldest, most quiescent molecular clouds.
Both isomers are highly polar, with measured dipole moments along their
$a$-inertial axes in excess of 4 D [4.15(15) D and 4.36(25) D, respectively;
(Sakaizumi et al., 1987)], and comparable to that of benzonitrile
($\mu_{a}=4.5152$ D; (Wohlfart et al., 2008)).

On the basis of highly accurate laboratory rest frequencies, and the first
data release (DR1) from a large-scale, high sensitivity spectral line survey,
GBT Observations of TMC-1: Hunting for Aromatic Molecules (GOTHAM),
predominately in the K (18–27 GHz) and Ka bands (26–40 GHz), a number of the
authors here recently reported the astronomical detection of 1-cyano-CPD using
spectral stacking and matched filtering techniques (McCarthy et al., 2020b).
From these observations, which represent ${\sim}$30% of the project goal, an
upper limit for 2-cyano-CPD relative to 1-cyano-CPD was estimated to be
roughly 1/3. Because the abundance ratio in the laboratory ranges from 1/2 to
1/4 (Sakaizumi et al., 1987; McCarthy et al., 2020a), it was unclear if
1-cyano-CPD is formed selectively in TMC-1 or if the apparent absence of
2-cyano-CPD is simply a question of sensitivity, given its somewhat lower
stability and consequently lower abundance. With the second data release (DR2)
and additional laboratory measurements, this ambiguity has now been resolved,
and in doing so a common formation for this isomeric pair is implicated.

##  2 Observations and Data Analysis

The observations in DR2 fold in new observations made between February 2018
and June 2020 on the Robert C. Byrd 100-m Green Bank Telescope in Green Bank,
West Virginia under project codes GBT18A-333, GBT18B-007, and GBT19A-047.
Although the frequency range of the present dataset is only slightly wider
than that covered in DR1 (McGuire et al., 2020b), it is considerably more
sensitive at some frequencies (McGuire et al., 2020a). The spectral coverage
of DR2 extends from 7.906 to 33.527 GHz (25.6 GHz bandwidth) with continuous
coverage between 22–33.5 GHz, at a uniform frequency resolution of 1.4 kHz
(0.05–0.01 km/s in velocity) and an RMS noise of ${\sim}2$–20 mK ($T_{A}*$)
across the spectrum.

As before, the target was the cyanopolyyne peak of TMC-1 at (J2000) $\alpha$ =
04h41m42.50s $\delta$ = +25∘41′26.8′′. The calibrator source for pointing and
focus observations was J0530+1331; focus and pointing offsets were performed
at the beginning of each observing session, and subsequently every 1 to 2 h,
depending on the weather; typical pointing convergence was $\lesssim$5′′.
Observations were performed using position switching (ON-OFF), in which the
target and the off position were observed in a sequential manner, each for 2
min. The off position was chosen to be 1∘ off target and was confirmed to be
clear of emission. Additionally data from project GBT17A-164 and GBT17A-434
have also been folded in the DR2. The observing strategy for this archival
data is outlined in McGuire et al. (2018), but it is very similar to that used
here. To ensure uniformity and consistency with the present data set, the
archival data were re-calibrated and re-reduced. Uncertainty due to flux
calibration is expected to be $\sim$20%, based on complementary VLA
observations of the flux-calibrator source J0530+1331 (McGuire et al., 2020c).

A detailed description of the data analysis and statistical methods, including
the procedure for spectral stacking and matched filtering, are presented
elsewhere (Loomis et al., 2020), so only a brief summary is provided here.
Briefly, many small frequency regions, each centered around a predicted
transition frequency of a target species, are extracted from the full dataset
of observations. Any window containing an obvious spectral feature (e.g.,
$>$5$\sigma$) is omitted so as to avoid any interlopers: this corresponds to
275 transitions for 1-cyano-CPD and 326 for 2-cyano-CPD, without interlopers
detected. A signal-to-noise weighted average of the spectra was then performed
based on the expected intensity of the line and the RMS noise of the
observations. Only transitions of the target species that have a predicted
flux $\geq 5$% of the strongest line are considered in our analysis. This
procedure is built into the molsim package (Lee & McGuire, 2020), which
performs the spectral simulation and wraps the affine-invariant MCMC
implementation of emcee (Foreman-Mackey et al., 2013) and posterior analysis
routines from ArviZ (Kumar et al., 2019).

Table 1: Spectroscopic constants of 1-cyano-CPD and 2-cyano-CPD. The fits were
performed with the Watson A-reduced Hamiltonian including quartic centrifugal
distortion. All values are given in MHz with 1$\sigma$ uncertainties in
parentheses. Values bracketed with [ ] were held fixed.

3  Parameter 1-cyano-CPD 2-cyano-CPD This work McCarthy et al. (2020b) This
work McCarthy et al. (2020b) $A$ $8352.981(10)$ $8352.98(2)$ $8235.592(14)$
$8235.66(4)$ $B$ $1904.2522(2)$ $1904.2514(3)$ $1902.0748(3)$ $1902.0718(2)$
$C$ $1565.3652(2)$ $1565.3659(3)$ $1559.6472(2)$ $1559.6502(2)$
$\Delta_{J}\times 10^{3}$ $0.0743(11)$ $0.0701(15)$ $0.0686(11)$ $0.0561(37)$
$\Delta_{JK}\times 10^{3}$ $2.354(8)$ $2.361(15)$ $2.287(21)$ $2.286(46)$
$\Delta_{K}\times 10^{3}$ $[0.17561]^{a}$ $[0.]$ $[0.32391]^{a}$ $[0.]$
$\delta_{J}\times 10^{3}$ $0.0133(5)$ $0.0120(11)$ $0.0134(6)$ $[0.]$
$\delta_{K}\times 10^{3}$ $1.48(9)$ $1.21(13)$ $1.10(9)$ $[0.]$
$\chi_{aa}(\mathrm{N})$ $-4.1810(11)$ $-4.1796(21)$ $-4.2429(13)$ $-4.234(6)$
$\chi_{bb}(\mathrm{N})$ $2.3016(14)$ $2.3052(26)$ $2.2475(16)$ $2.236(7)$
$N_{\text{lines}}$ b $154$ $68$ $110$ $38$ $(J,K_{a})_{\text{max}}$ $(11,3)$
$(9,3)$ $(10,2)$ $(5,2)$ a Calculated at the $\omega$B97X-D/cc-pVQZ level of
theory. b The number of hyperfine-resolved transitions included in the fit.

Given that the 1-cyano-CPD isomer was characterized in our earlier work, we
used the posterior distributions obtained in McCarthy et al. (2020b) as priors
for both 1-cyano-CPD and 2-cyano-CPD MCMC simulations here—effectively
refining the previous model with the new experimental and observational data.
The model space comprises 14 parameters corresponding to four known velocity
components within TMC-1 (Dobashi et al., 2018, 2019), each having an
independent source size [SS], column density ([$N_{T}$], and radial velocity
[$v_{lsr}$], while a common excitation temperature [$T_{ex}$] and linewidth
[$dv$] are assumed. The parameters are used to simulate the expected flux in a
forward model, taking into account beam dilution and optical depth effects,
with the MCMC sampling guided by the observed spectra. Finally, to determine
an overall significance of a detection, the model spectra are stacked using
identical weights, and that stacked model is then used as a matched filter
that is cross-correlated with the stacked observations. The resulting response
spectrum provides a lower limit on the statistical significance to the
detections.

##  3 Laboratory Measurements

The astronomical detection of 1-cyano-CPD was based on laboratory measurements
made using a cavity-enhanced Fourier transform microwave spectrometer in which
this molecule and many others (McCarthy et al., 2020a, b) were produced in
discharge of benzene and molecular nitrogen. To improve the accuracy of the
spectroscopic constants of 1-cyano-CPD and measure a comparable number of
transitions for 2-cyano-CPD, a discharge of dicyclopentadiene (the Diels-Alder
dimer of CPD) and acetonitrile was used here. This precursor combination
yielded nearly a ten-fold increase in line intensity for both isomers relative
to our earlier work. For 1-cyano-CPD, the number of hyperfine components in
the laboratory data set approximately doubled, while a three-fold increase was
achieved for 2-cyano-CPD. In doing so, the frequency range of the measurements
increased commensurately: from 30 to 36 GHz for 1-cyano-CPD, and from 19 GHz
to 33 GHz for 2-cyano-CPD. The best-fit spectroscopic constants for both
species—the most complete summary of the rotational data yet—are summarized in
Table 1.

As part of these measurements, improved SNR for both isomers enabled
transitions between higher $J$ and $K_{a}$ levels to be observed. Inclusion of
these weaker transitions proportionally increases the rotational partition
functions, and alters somewhat the column densities derived for both species.
At 300 K, the new values for $Q_{\mathrm{rot}}$ are approximately 1.4 times
larger than in McCarthy et al. (2020b), resulting in a substantially lower
column density, and one outside the 2$\sigma$ uncertainties reported in our
previous analysis.

Figure 2: Velocity stacked and matched filter spectra of 1-cyano-CPD and
2-cyano-CPD. (_Left_) The stacked spectra from the GOTHAM DR2 data are display
in blue, overlaid with the expected line profile in red from our MCMC fit to
the data. The signal-to-noise ratio is on a per-channel basis. (_Right_)
Impulse response functions of the stacked spectra of same molecules using the
simulated line profiles as matched filters. The intensity scales are the
signal-to-noise ratios of the response functions when centered at a given
velocity. The values denoted in each figure indicate the peak of the impulse
response functions which provide a minimum significance for the detection.

##  4 Results and Discussion

Shown in Fig. 2 are stacked spectra and the impulse responses for the two
cyanocyclopentadiene isomers that have been derived from the DR2 data.
Additional results of the MCMC results can be found in the Appendix. Relative
to the DR1 results, these additional observations improve the detection
significance of 1-cyano-CPD by a factor of 1.8 (from 5.8$\sigma$ to
10.7$\sigma$), and more importantly, provide compelling evidence for 2-cyano-
CPD. As the visual representations of the statistical analysis illustrate, we
can now conclude with good confidence that both isomers are present in TMC-1.
Furthermore, albeit at low signal-to-noise ratio (SNR), evidence for several
individual rotational lines of 1-cyano-CPD as shown in Fig 3; these correspond
to the strongest transitions at ${\sim}8$ K in regions of our spectra with
varying SNR.

Tables A1 and A2 summarize the best-fit values from our MCMC analysis. In
agreement with prior work (Dobashi et al., 2018, 2019), four distinct velocity
components in the source with nominally independent four column densities and
four source sizes were included in our model. In comparison with our DR1
analysis, the higher frequency coverage in DR2 provides better constraints on
the source sizes and in turn the derived column densities. Most of the derived
quantities are similar in magnitude to those reported previously, and, as
before, the source sizes remain poorly constrained. Most importantly, an
abundance ratio of $4.4(6)$ (1$\sigma$ uncertainty) has been determined for
the two isomers; we note, however that this value treats the two models as
statistically independent, and a proper estimate would require explicit
treatment of the column density covariances between the two molecules.

Figure 3: Four rotational transitions of 1-cyano-CPD observed towards TMC-1
using the 100-m GBT telescope. The overlapping red trace and shaded regions
represent the simulated flux based on mean parameters derived from the MCMC
posterior. The corresponding asymmetric top ($J_{K_{a},K_{c}}$) quantum
numbers for each transition are shown above each feature. The dashed line
indicates the nominal 5.8 km/s source velocity.

The astronomical detection of 2-cyano-CPD with a marginally similar abundance
to 1-cyano-CPD strongly suggests both isomers are formed via a common pathway.
The simplest and most direct route involves the reaction of cyclopentadiene
with CN radical. Although apparently not examined experimentally, theoretical
calculations (McCarthy et al., 2020b) at the G3//B3LYP level of theory predict
both 1-cyano-CPD and 2-cyano-CPD are formed exothermically and barrierlessly,
irrespective of the small difference in their stabilities. This reaction is
analogous to benzonitrile formation from benzene and CN, which has been
extensively studied by multiple experimental (Balucani et al., 2000; Trevitt
et al., 2010; Lee et al., 2019; Cooke et al., 2020) and theoretical techniques
(Woon, 2006; Lee et al., 2019), and is known to occur rapidly at low
temperature, implying the possibility of in situ formation of cyanide
derivatives of hydrocarbons under the cold, dark conditions of TMC-1. By
analogy, the same logic implies the presence of cyclopentadiene in TMC-1,
although the small permanent dipole moment [$\mu_{b}=0.420$ D; (Laurie, 1956;
Damiani et al., 1976)] of this hydrocarbon ring will make its direct detection
very challenging in the radio band. It is also worth noting the abundance of
the two cyanocyclopentadienes taken together close to that of benzonitrile
[$1.60\times 10^{12}$ cm-2; Burkhardt et al. (2020)]. Since both the five and
six-membered rings likely form via reaction with a common precursor, the CN
radical, and the cyanation reactions are highly exothermic and barrierless,
this ratio may reflect the nascent hydrocarbon abundances in TMC-1. Although
there is no obvious a priori expectation of the benzene/cyclopentadiene
abundance ratio, the former is aromatic while the latter is not, and thus
might be expected to more abundant based on thermodynamic stability.

In a more general sense, the present work further highlights that carbon
chemistry of considerable richness and complexity lies just below the noise
floor of previous spectral line surveys toward this well-known primordial
molecular cloud. Despite discoveries of monocyclic and even bicyclic rings in
rapid succession, this work has raised many more questions than it has helped
answer. Nevertheless, it serves to illustrate that there is potentially much
more to be learned about a well-studied and until recently a seemingly well-
understood source. In this context, it would be disappointing if still other
functionalized rings and potentially their precursors are not eventually found
with sustained effort.

##  5 Data access & code

Data used for the MCMC analysis can be found in the DataVerse entry (GOTHAM,
2020). The code used to perform the analysis is part of the molsim open-source
package; an archival version of the code can be accessed at Lee & McGuire
(2020).

M.C.M, K.L.K.L., and P.B.C. acknowledge financial support from NSF grants
AST-1908576, AST-1615847, and NASA grant 80NSSC18K0396. A.M.B. acknowledges
support from the Smithsonian Institution as a Submillimeter Array (SMA)
Fellow. S.B.C. and M.A.C. were supported by the NASA Astrobiology Institute
through the Goddard Center for Astrobiology. C.X. is a Grote Reber Fellow, and
support for this work was provided by the NSF through the Grote Reber
Fellowship Program administered by Associated Universities, Inc./National
Radio Astronomy Observatory and the Virginia Space Grant Consortium. The
National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. The Green Bank Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc.

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##  Appendix A MCMC results

The following summarizes the posteriors obtained for 1-cyano-CPD and 2-cyano-
CPD from the MCMC modeling. The data comprises corner plots and statistics
that visualizes and quantifies the sampling results. Corner plots are
interpreted in two ways: the off-diagonal contours show the covariance between
model parameters, while the diagonal traces are empirical cumulative
distribution function (ECDF) plots. The latter indicates the integrated
marginalized likelihood of a given parameter—in other words, the parameter
space comprised by the posterior.

###  A.1 1-cyano-CPD

Figure A1 shows the posterior traces for the MCMC modeling analysis of
1-cyano-CPD. These simulations indicate detections in all four components,
with the source sizes very poorly constrained and covariant with the column
densities. Table A1 provides summary statistics of the marginalized
posteriors.

Figure A1: Corner plot for 1-cyano-CPD The diagonal traces correspond to ECDF
plots, and off-diagonal plots show the kernel density covariance between model
parameters. In the former, lines represent the 25${}^{\mbox{th}}$,
50${}^{\mbox{th}}$, and 75${}^{\mbox{th}}$ percentiles respectively. The
length scale for the kernel density plots is chosen with Scott’s rule. Table
A1: 1-cyano-CPD best-fit parameters from MCMC analysis.

Component | $v_{lsr}$ | Size | $N_{T}^{\dagger}$ | $T_{ex}$ | $\Delta V$
---|---|---|---|---|---
(km s-1) | (′′) | (1011 cm-2) | (K) | (km s-1)
C1 | $5.617^{+0.015}_{-0.015}$ | $132^{+67}_{-66}$ | $1.03^{+0.20}_{-0.20}$ | $6.00^{+0.03}_{-0.03}$ | $0.119^{+0.009}_{-0.010}$
C2 | $5.795^{+0.022}_{-0.023}$ | $55^{+30}_{-30}$ | $1.57^{+1.03}_{-0.86}$
C3 | $5.869^{+0.033}_{-0.033}$ | $14^{+5}_{-4}$ | $4.69^{+1.60}_{-1.62}$
C4 | $6.015^{+0.016}_{-0.017}$ | $232^{+153}_{-161}$ | $0.92^{+0.23}_{-0.27}$ |  |
$N_{T}$ (Total)†† |  $8.27^{+0.90}_{-1.00}\times 10^{11}$ cm-2

Note – The quoted uncertainties represent 95% highest posterior density.
†Column density values are highly covariant with the derived source sizes.
††Total column density derived from combining posterior column densities of
each component. The uncertainty corresponds to the 95% highest joint posterior
density.

###  A.2 2-cyano-CPD

Figure A2 shows the posterior traces for the MCMC modeling analysis of
2-cyano-CPD. These simulations suggest detections in components #1 and #4, and
components #2 and #3 are degenerate with the source sizes effectively poorly
constrained in the all cases. Table A2 provides summary statistics of the
marginalized posteriors.

Figure A2: Corner plot for 2-cyano-CPD The diagonal traces correspond to ECDF
plots, and off-diagonal plots show the kernel density covariance between model
parameters. In the former, lines represent the 25${}^{\mbox{th}}$,
50${}^{\mbox{th}}$, and 75${}^{\mbox{th}}$ percentiles respectively. The
length scale for the kernel density plots is chosen with Scott’s rule. Table
A2: 2-cyano-CPD best-fit parameters from MCMC analysis.

Component | $v_{lsr}$ | Size | $N_{T}^{\dagger}$ | $T_{ex}$ | $\Delta V$
---|---|---|---|---|---
(km s-1) | (′′) | (1011 cm-2) | (K) | (km s-1)
C1 | $5.612^{+0.060}_{-0.050}$ | $110^{+71}_{-90}$ | $0.27^{+0.21}_{-0.24}$ | $6.00^{+0.03}_{-0.03}$ | $0.122^{+0.010}_{-0.010}$
C2 | $5.754^{+0.040}_{-0.039}$ | $67^{+31}_{-32}$ | $0.64^{+0.54}_{-0.64}$
C3 | $5.836^{+0.126}_{-0.056}$ | $249^{+140}_{-129}$ | $0.54^{+0.44}_{-0.51}$
C4 | $6.054^{+0.028}_{-0.028}$ | $239^{+140}_{-135}$ | $0.48^{+0.15}_{-0.14}$ |  |
$N_{T}$ (Total)†† |  $1.89^{+0.18}_{-0.15}\times 10^{11}$ cm-2

Note – The quoted uncertainties represent the 95% highest posterior density.
†Column density values are highly covariant with the derived source sizes.
††Total column density is given as the mean combined posterior column
densities of each component. The uncertainty corresponds to the 95% highest
joint posterior density.