# The set of ratios of derangements to permutations in digraphs is dense in
$\displaystyle[0,1/2]$.

Bethany Austhof Department of Mathematics, Statistics, and Computer Science,
University of Illinois Chicago, Chicago, IL, USA<EMAIL_ADDRESS>Patrick
Bennett Department of Mathematics, Western Michigan University, Kalamazoo, MI,
USA<EMAIL_ADDRESS>Research supported in part by Simons Foundation
Grant #426894. Nick Christo  Department of Mathematics, Statistics, and
Computer Science, University of Illinois Chicago, Chicago, IL, USA
<EMAIL_ADDRESS>
###### Abstract

A permutation in a digraph $\displaystyle G=(V,E)$ is a bijection
$\displaystyle f:V\rightarrow V$ such that for all $\displaystyle v\in V$ we
either have that $\displaystyle f$ fixes $\displaystyle v$ or
$\displaystyle(v,f(v))\in E$. A derangement in $\displaystyle G$ is a
permutation that does not fix any vertex. In [1] it is proved that in any
digraph, the ratio of derangements to permutations is at most $\displaystyle
1/2$. Answering a question posed in [1], we show that the set of possible
ratios of derangements to permutations in digraphs is dense in the interval
$\displaystyle[0,1/2]$.

##  1 Introduction

A permutation in a digraph $\displaystyle G=(V,E)$ is a bijection
$\displaystyle f:V\rightarrow V$ such that for all $\displaystyle v\in V$ we
either have that $\displaystyle f$ fixes $\displaystyle v$ or
$\displaystyle(v,f(v))\in E$. A derangement in $\displaystyle G$ is a
permutation that does not fix any vertex. We define the parameter
$\displaystyle(d/p)_{G}$ to be the ratio of derangements to permutations in
$\displaystyle G$.

Bucic, Devlin, Hendon, Horne and Lund [1] showed that
$\displaystyle(d/p)_{G}\leq 1/2$ for all digraphs $\displaystyle G$, with
equality if and only if $\displaystyle G$ is a directed cycle. They also gave
a construction (the blow-up of a directed cycle) that can achieve a ratio
arbitrarily close to but not equal to $\displaystyle 1/2$. Let $\displaystyle
S=\\{(d/p)_{G}\;:\;G\;\;\text{is a digraph}\\}$ be a set of values arising as
a ratio $\displaystyle(d/p).$ In [1] they analyzed the ratio
$\displaystyle(d/p)_{G}$ for the random graph $\displaystyle G=G(n,m)$, and as
a corollary of this analysis they showed that $\displaystyle S$ is dense in
$\displaystyle[0,1/e]$. This corollary follows from two facts:
$\displaystyle(d/p)_{G}$ is concentrated around its mean, and by choosing a
suitable value of $\displaystyle m$ one can make the expected ratio
$\displaystyle(d/p)_{G}$ close to any given value in $\displaystyle[0,1/e]$.
At the end of the paper [1] they ask whether $\displaystyle S$ is dense in
$\displaystyle[0,1/2]$. Our main theorem, below, answers this question in the
positive.

######  Theorem 1.

The set of possible ratios of derangements to permutations in digraphs is
dense in $\displaystyle[0,1/2]$.

The construction we use, described in more detail later, is a random subgraph
of the blow-up of a directed cycle. The main part of the proof is an
application of the second moment method (see [2], for example, for an
introduction to the method) to show that the number of derangements and
permutations are concentrated around their expectations.

##  2 Proof of Theorem 1

###  2.1 Outline

First we outline the proof. Suppose we are given a fixed real number
$\displaystyle r\in[0,1/2]$. We will show that there exists a sequence of
digraphs $\displaystyle G_{k}$ such that the ratio of derangements to
permutations in $\displaystyle G_{k}$ is $\displaystyle r+o(1)$ as
$\displaystyle k\rightarrow\infty$ (which proves Theorem 1). If $\displaystyle
r=0$ or $\displaystyle 1/2$ this is trivial. Indeed, for $\displaystyle r=0$
observe that a digraph with one vertex and no edges has no derangements and
one permutation, and for $\displaystyle r=1/2$ observe that any directed cycle
has one derangement and two permutations. So we assume $\displaystyle
0<r<1/2$.

Our construction is as follows. As defined in [1], let the digraph
$\displaystyle D_{k,\ell}$ where $\displaystyle k\geq 1$ and
$\displaystyle\ell\geq 2$ have vertices $\displaystyle v_{ij}$ for
$\displaystyle i\in[k]$ and $\displaystyle j\in[\ell]$ such that
$\displaystyle(v_{ij},v_{lm})\in E(D_{k,\ell})$ if and only if $\displaystyle
m=j+1\,\text{mod}\,l$. In other words, $\displaystyle D_{k,\ell}$ is the blow-
up of a directed $\displaystyle\ell$-cycle where each vertex is expanded to a
set of $\displaystyle k$ vertices. We let $\displaystyle
V_{i}=\\{v_{ij}:j\in[\ell]\\}$. As was shown in [1], the number of
derangements on $\displaystyle D_{k,\ell}$ is $\displaystyle(k!)^{\ell}$ and
the number of permutations on $\displaystyle D_{k,\ell}$ is
$\displaystyle\sum_{i=0}^{k}\left(\binom{k}{i}(k-i)!\right)^{\ell}$. Hence,
$\displaystyle(d/p)_{k,\ell}=\left(\sum_{i=0}^{k}\left(\frac{1}{i!}\right)^{\ell}\right)^{-1}$
can be made arbitrarily close to 1/2 by choosing $\displaystyle\ell$ large
enough (even for large $\displaystyle k$). This construction yields a graph
for which the ratio of derangements to permutations is arbitrarily close to
1/2 but not exactly 1/2. We will also use this construction, but we will
randomly remove some edges. By taking a random subraph we can “interpolate”
between $\displaystyle D_{k,\ell}$ (a dense digraph whose ratio of
derangements to permutations is close to $\displaystyle 1/2$) and a sparse
random digraph (whose ratio is $\displaystyle 0$).

In this paper all asymptotics are as $\displaystyle k\rightarrow\infty$.
$\displaystyle\ell$ is treated as fixed. We use standard big-O, little-o and
$\displaystyle\Omega$ notation. We write $\displaystyle x\sim y$ if
$\displaystyle x=(1+o(1))y$. All logarithms are base $\displaystyle e$.

###  2.2 Proof details

Let the random graph $\displaystyle G_{k,\ell}(m)$ be chosen uniformly from
among all subgraphs of $\displaystyle D_{k,\ell}$ with $\displaystyle m$
edges. We will fix some $\displaystyle p,\ell$ and let $\displaystyle
m=pk^{2}\ell$ (so $\displaystyle p$ is the probability that any particular
edge of $\displaystyle D_{k,\ell}$ becomes an edge of $\displaystyle
G_{k,\ell}$). Let the random variables $\displaystyle X,Y$ be the number of
derangements and permutations in $\displaystyle G_{k,\ell}(m)$ respectively.
Let $\displaystyle\mathcal{D},\mathcal{P}$ be the collection of all possible
derangements and permutations on $\displaystyle D_{k,\ell}(m)$.

####  2.2.1 First moments of $\displaystyle X,Y$

We have

$\displaystyle\displaystyle\mathbb{E}[X]$
$\displaystyle\displaystyle=\sum_{D\in\mathcal{D}}\mathbb{P}[D\subseteq
G_{k,\ell}]=(k!)^{\ell}\frac{\binom{k^{2}\ell-k\ell}{m-k\ell}}{\binom{k^{2}\ell}{m}}$
$\displaystyle\displaystyle=(k!)^{\ell}\left(\frac{m}{k^{2}\ell}\right)^{k\ell}\exp\left\\{\frac{k^{2}\ell^{2}}{2}\left(\frac{1}{k^{2}\ell}-\frac{1}{m}\right)+O\left(\frac{k^{3}}{m^{2}}+\frac{k}{m}\right)\right\\}$
$\displaystyle\displaystyle\sim(k!)^{\ell}p^{k\ell}\exp\left\\{\frac{\ell}{2}\left(1-\frac{1}{p}\right)\right\\}$
(2.1)

where on the second line we have used the following fact:

######  Fact 2.1.

$\displaystyle\frac{\binom{a-x}{b-x}}{\binom{a}{b}}=\frac{(b)_{x}}{(a)_{x}}=\left(\frac{b}{a}\right)^{x}\exp\left\\{\frac{x^{2}}{2}\left(\frac{1}{a}-\frac{1}{b}\right)+O\left(\frac{x^{3}}{b^{2}}+\frac{x}{b}\right)\right\\}.$

For completeness we include the proof although it is well-known.

###### Proof.

$\displaystyle\displaystyle\frac{(b)_{x}}{(a)_{x}}$
$\displaystyle\displaystyle=\left(\frac{b}{a}\right)^{x}\cdot\frac{1\left(1-\frac{1}{b}\right)\left(1-\frac{2}{b}\right)\cdots\left(1-\frac{x-1}{b}\right)}{1\left(1-\frac{1}{a}\right)\left(1-\frac{2}{a}\right)\cdots\left(1-\frac{x-1}{a}\right)}$
$\displaystyle\displaystyle=\left(\frac{b}{a}\right)^{x}\cdot\exp\left\\{\sum_{i=0}^{x-1}\left[{\ln\left(1-\frac{i}{b}\right)-\ln\left(1-\frac{i}{a}\right)}\right]\right\\}$
$\displaystyle\displaystyle=\left(\frac{b}{a}\right)^{x}\cdot\exp\left\\{\sum_{i=0}^{x-1}\left[{-\frac{i}{b}+\frac{i}{a}+O\left(\frac{i^{2}}{a^{2}}+\frac{i^{2}}{b^{2}}\right)}\right]\right\\}$
$\displaystyle\displaystyle=\left(\frac{b}{a}\right)^{x}\cdot\exp\left\\{\frac{x(x-1)}{2}\left(\frac{1}{a}-\frac{1}{b}\right)+O\left(\frac{x^{3}}{b^{2}}\right)\right\\}$
$\displaystyle\displaystyle=\left(\frac{b}{a}\right)^{x}\exp\left\\{\frac{x^{2}}{2}\left(\frac{1}{a}-\frac{1}{b}\right)+O\left(\frac{x^{3}}{b^{2}}+\frac{x}{b}\right)\right\\}.$

∎

Before we calculate $\displaystyle\mathbb{E}[Y]$ we introduce a function
$\displaystyle f_{\ell}(x)$. For any integer $\displaystyle\ell\geq 1$, let

$f_{\ell}(x):=\sum_{i=0}^{\infty}\frac{x^{i\ell}}{(i!)^{\ell}}.$ (2.2)

Note that the above power series for $\displaystyle f_{\ell}(x)$ converges for
all $\displaystyle x$ and therefore in particular each $\displaystyle
f_{\ell}$ is continuous in $\displaystyle x$. We have

$\displaystyle\displaystyle\mathbb{E}[Y]$
$\displaystyle\displaystyle=\sum_{P\in\mathcal{P}}\mathbb{P}[P\subseteq
G_{k,\ell}]=\sum_{i=0}^{k}\left(\binom{k}{i}(k-i)!\right)^{\ell}\frac{\binom{k^{2}\ell-(k-i)\ell}{m-(k-i)\ell}}{\binom{k^{2}\ell}{m}}$
$\displaystyle\displaystyle=\sum_{i=0}^{k}\left(\frac{k!}{i!}\right)^{\ell}\left(\frac{m}{k^{2}\ell}\right)^{(k-i)\ell}\exp\left\\{\frac{(k-i)^{2}\ell^{2}}{2}\left(\frac{1}{k^{2}\ell}-\frac{1}{m}\right)+O\left(\frac{k^{3}}{m^{2}}+\frac{k}{m}\right)\right\\}$
$\displaystyle\displaystyle=(k!)^{\ell}p^{k\ell}\sum_{i=0}^{k}\left(\frac{1}{i!}\right)^{\ell}p^{-i\ell}\exp\left\\{\frac{\ell}{2}\left(1-\frac{1}{p}\right)+O\left(\frac{i+1}{k}\right)\right\\}.$
(2.3)

We split the above sum into two ranges of $\displaystyle i$. Note that for
$\displaystyle 0\leq i\leq\sqrt{k}$ we have
$\displaystyle\exp\left\\{O\left(\frac{i+1}{k}\right)\right\\}=1+O\left(\frac{1}{\sqrt{k}}\right)$,
while for $\displaystyle\sqrt{k}\leq i\leq k$ we have
$\displaystyle\exp\left\\{O\left(\frac{i+1}{k}\right)\right\\}=O(1)$. Thus
line (2.3) becomes

$(k!)^{\ell}p^{k\ell}\left[{\left(1+O\left(\frac{1}{\sqrt{k}}\right)\right)\exp\left\\{\frac{\ell}{2}\left(1-\frac{1}{p}\right)\right\\}\sum_{0\leq
i\leq\sqrt{k}}\left(\frac{1}{i!}\right)^{\ell}p^{-i\ell}+O(1)\sum_{\sqrt{k}<i\leq
k}\left(\frac{1}{i!}\right)^{\ell}p^{-i\ell}}\right].$ (2.4)

As $\displaystyle k\rightarrow\infty$ we have

$\displaystyle\sum_{0\leq
i\leq\sqrt{k}}\left(\frac{1}{i!}\right)^{\ell}p^{-i\ell}\rightarrow
f_{\ell}(1/p),$

and

$\displaystyle\sum_{\sqrt{k}<i\leq
k}\left(\frac{1}{i!}\right)^{\ell}p^{-i\ell}\leq\sum_{i=\sqrt{k}}^{\infty}\left(\frac{1}{i!}\right)^{\ell}p^{-i\ell}=o(1)$

since the latter is the tail of a convergent series. Thus, returning to our
estimate of $\displaystyle\mathbb{E}[Y]$ on line (2.4), we have

$\mathbb{E}[Y]\sim(k!)^{\ell}p^{k\ell}\exp\left\\{\frac{\ell}{2}\left(1-\frac{1}{p}\right)\right\\}f_{\ell}(1/p).$
(2.5)

####  2.2.2 Choosing $\displaystyle p,\ell$

Now that we know $\displaystyle\mathbb{E}[X],\mathbb{E}[Y]$ we will choose
$\displaystyle p,\ell$ to make sure that the ratio of
$\displaystyle\mathbb{E}[X]$ to $\displaystyle\mathbb{E}[Y]$ is close to
$\displaystyle r$. Using lines (2.1) and (2.5) we have

$\displaystyle\frac{\mathbb{E}[X]}{\mathbb{E}[Y]}\sim\frac{1}{f_{\ell}\left(\frac{1}{p}\right)},$

so we would like to choose $\displaystyle\ell$ and $\displaystyle 0<p<1$ so
that $\displaystyle f_{\ell}(1/p)=1/r$. We have

$\displaystyle\lim_{x\rightarrow\infty}f_{\ell}(x)=\infty,\qquad
f_{\ell}(1)=\sum_{i=0}^{k}\left(\frac{1}{i!}\right)^{\ell}=1+1+\frac{1}{2^{\ell}}+\frac{1}{6^{\ell}}+\frac{1}{24^{\ell}}+\ldots.$

Note that we can make $\displaystyle f_{\ell}(1)$ arbitrarily close to 2 by
taking $\displaystyle\ell$ large. Indeed, we have $\displaystyle
f_{\ell}(1)\geq 2$ and

$\displaystyle\displaystyle f_{\ell}(1)=2+\sum_{i\geq
2}\left(\frac{1}{i!}\right)^{\ell}$ $\displaystyle\displaystyle\leq
2+\sum_{i\geq 2}\left(\frac{1}{2^{i-1}}\right)^{\ell}=2+\frac{1}{2^{\ell}-1}.$

Since $\displaystyle r<1/2$, we can choose $\displaystyle\ell$ so that
$\displaystyle f_{\ell}(1)<1/r$. Then by the intermediate value theorem there
is some $\displaystyle x\in(1,\infty)$ such that $\displaystyle
f_{\ell}(x)=1/r$. We choose $\displaystyle p$ to be the value $\displaystyle
1/x$, so $\displaystyle 0<p<1$ and $\displaystyle f_{\ell}(1/p)=1/r$. So we
view $\displaystyle\ell$ and $\displaystyle p$ as constants determined
entirely by $\displaystyle r$.

####  2.2.3 Second moments of $\displaystyle X,Y$

In this section we show that
$\displaystyle\mathbb{E}[X^{2}]\sim\mathbb{E}[X]^{2}$ and
$\displaystyle\mathbb{E}[Y^{2}]\sim\mathbb{E}[Y]^{2}$. This will complete the
proof, since then by the second moment method we have that

$\displaystyle\frac{X}{Y}\sim\frac{\mathbb{E}[X]}{\mathbb{E}[Y]}\sim\frac{1}{f_{\ell}(1/p)}=r.$

with probability approaching 1 as $\displaystyle k$ goes to infinity.

To help us estimate $\displaystyle\mathbb{E}[X^{2}],\mathbb{E}[Y^{2}]$ we will
find the function $\displaystyle h(a,b)$ (defined below) useful. Suppose we
have some fixed matching $\displaystyle B$ of $\displaystyle b$ many edges in
the graph $\displaystyle K_{a,a}$. Then by inclusion-exclusion the number of
perfect matchings that do not have any edges from $\displaystyle B$ is

$\displaystyle h(a,b):=\sum_{w=0}^{b}(-1)^{w}\binom{b}{w}(a-w)!.$

Note that we always have $\displaystyle h(a,b)\leq a!$. We will now observe
that, roughly speaking, $\displaystyle h(a,b)\approx\frac{a!}{e}$ whenever
$\displaystyle b\approx a\rightarrow\infty$. More formally we have the
following

######  Fact 2.2.

Suppose $\displaystyle a-a^{1/10}\leq b\leq a$. Then we have

$\displaystyle h(a,b)=\left(1+O(a^{-4/5})\right)\frac{a!}{e}$

as $\displaystyle a\rightarrow\infty$

###### Proof.

We have

$\displaystyle\displaystyle h(a,b)=\sum_{0\leq w\leq
b}(-1)^{w}\binom{b}{w}(a-w)!=a!\sum_{0\leq w\leq
b}\frac{(-1)^{w}}{w!}\frac{(b)_{w}}{(a)_{w}}.$ (2.6)

Now, for $\displaystyle 0\leq w\leq a^{1/10}$ we have by Fact 2.1 that

$\displaystyle\displaystyle\frac{(b)_{w}}{(a)_{w}}$
$\displaystyle\displaystyle=\left(\frac{b}{a}\right)^{w}\exp\left\\{\frac{w^{2}}{2}\left(\frac{1}{a}-\frac{1}{b}\right)+O\left(\frac{w^{3}}{b^{2}}+\frac{w}{b}\right)\right\\}$
$\displaystyle\displaystyle=\left(1+O\left(a^{-9/10}\right)\right)^{O\left(a^{1/10}\right)}\exp\left\\{O(a^{-4/5}\right\\}=1+O(a^{-4/5}).$

Meanwhile for $\displaystyle w\geq a^{1/10}$ we have that the corresponding
term in line (2.6) has absolute value

$\displaystyle\displaystyle\frac{1}{w!}\frac{(b)_{w}}{(a)_{w}}\leq\frac{1}{(a^{1/10})!}=\exp\left\\{-\Omega\left(a^{1/10}\log
a\right)\right\\}$

by Stirling’s approximation. Thus, the sum of all such terms in line (2.6) is
at most

$\displaystyle b\exp\left\\{-\Omega\left(a^{1/10}\log
a\right)\right\\}=O(a^{-4/5})$

(this bound is quite comfortable). By the Alternating Series Test we have that

$\displaystyle\sum_{0\leq w\leq
a^{1/10}}\frac{(-1)^{w}}{w!}=\frac{1}{e}+O\left(\frac{1}{(a^{1/10})!}\right)=\frac{1}{e}+O(a^{-4/5}).$

Breaking up the sum for $\displaystyle h(a,b)$ we have

$\displaystyle\displaystyle h(a,b)$
$\displaystyle\displaystyle=a!\left[{\sum_{0\leq w\leq
a^{1/10}}\frac{(-1)^{w}}{w!}\frac{(b)_{w}}{(a)_{w}}+\sum_{a^{1/10}<w\leq
b}\frac{(-1)^{w}}{w!}\frac{(b)_{w}}{(a)_{w}}}\right]$
$\displaystyle\displaystyle=a!\left[{\left(1+O(a^{-4/5})\right)\sum_{0\leq
w\leq a^{1/10}}\frac{(-1)^{w}}{w!}+O(a^{-4/5})}\right]$
$\displaystyle\displaystyle=\left(1+O(a^{-4/5})\right)\frac{a!}{e}.$

∎

We find that

$\displaystyle\displaystyle\mathbb{E}[X^{2}]$
$\displaystyle\displaystyle=\sum_{D,D^{\prime}\in\mathcal{D}}\mathbb{P}[D,D^{\prime}\subseteq
G_{k\ell}]=(k!)^{\ell}\sum_{D^{\prime}\in\mathcal{D}}\mathbb{P}[D_{0},D^{\prime}\subseteq
M]$
$\displaystyle\displaystyle=(k!)^{\ell}\sum_{b=0}^{k\ell}\left[\frac{\binom{k^{2}\ell-(2k\ell-b)}{m-(2k\ell-b)}}{\binom{k^{2}\ell}{m}}\sum_{\vec{b}\in
S_{b}}\prod_{c=1}^{\ell}\binom{k}{b_{c}}h(k-b_{c},k-b_{c})\right].$ (2.7)

where in the inner sum, $\displaystyle S_{b}$ is the set of
$\displaystyle\ell$-dimensional vectors $\displaystyle\vec{b}=(b_{1},\ldots
b_{\ell})$ whose components are nonnegative integers summing to $\displaystyle
b$.

By 2.1, if $\displaystyle b\leq k^{1/10}$ then we have

$\displaystyle\displaystyle\frac{\binom{k^{2}\ell-(2k\ell-b)}{m-(2k\ell-b)}}{\binom{k^{2}\ell}{m}}$
$\displaystyle\displaystyle=p^{2k\ell-b}\exp\left\\{\frac{(2k\ell-b)^{2}}{2}\left(\frac{1}{k^{2}\ell}-\frac{1}{m}\right)+O\left(\frac{k^{3}}{m^{2}}+\frac{k}{m}\right)\right\\}$
$\displaystyle\displaystyle=\left(1+O(k^{-9/10})\right)p^{2k\ell-b}\exp\left\\{2\ell\left(1-\frac{1}{p}\right)\right\\}$

and by Fact 2.2 we have $\displaystyle
h(k-b_{c},k-b_{c})=\left(1+O(k^{-4/5})\right)\frac{(k-b_{c})!}{e}$. Therefore
the term corresponding to $\displaystyle b$ in (2.7) is

$\displaystyle\displaystyle\frac{\binom{k^{2}\ell-(2k\ell-b)}{m-(2k\ell-b)}}{\binom{k^{2}\ell}{m}}\sum_{\vec{b}\in
S_{b}}\prod_{c=1}^{\ell}$
$\displaystyle\displaystyle\binom{k}{b_{c}}h(k-b_{c},k-b_{c})$
$\displaystyle\displaystyle=\left(1+O(k^{-4/5})\right)p^{2k\ell-b}\exp\left\\{2\ell\left(1-\frac{1}{p}\right)\right\\}\sum_{\vec{b}\in
S_{b}}\prod_{c=1}^{\ell}\binom{k}{b_{c}}\frac{(k-b_{c})!}{e}$
$\displaystyle\displaystyle=\left(1+O(k^{-4/5})\right)(k!)^{\ell}p^{2k\ell-b}\exp\left\\{2\ell\left(1-\frac{1}{p}\right)-\ell\right\\}\sum_{\vec{b}\in
S_{b}}\prod_{c=1}^{\ell}\frac{1}{b_{c}!}$
$\displaystyle\displaystyle=\left(1+O(k^{-4/5})\right)(k!)^{\ell}p^{2k\ell-b}\exp\left\\{\ell\left(1-\frac{2}{p}\right)\right\\}\frac{\ell^{b}}{b!}$

where on the last line we used the multinomial formula. Meanwhile if
$\displaystyle b\geq k^{1/10}$ then the term corresponding to $\displaystyle
b$ in (2.7) is

$\displaystyle\displaystyle\frac{\binom{k^{2}\ell-(2k\ell-b)}{m-(2k\ell-b)}}{\binom{k^{2}\ell}{m}}\sum_{\vec{b}\in
S_{b}}\prod_{c=1}^{\ell}$
$\displaystyle\displaystyle\binom{k}{b_{c}}h(k-b_{c},k-b_{c})\leq\sum_{\vec{b}\in
S_{b}}\prod_{c=1}^{\ell}\binom{k}{b_{c}}(k-b_{c})!$
$\displaystyle\displaystyle=(k!)^{\ell}\frac{\ell^{b}}{b!}=(k!)^{\ell}\cdot\exp\left\\{-\Omega\left(k^{1/10}\log
k\right)\right\\}$

and so the sum of all terms in (2.7) with $\displaystyle b\geq k^{1/10}$ is at
most

$\displaystyle k\ell\cdot(k!)^{\ell}\cdot\exp\left\\{-\Omega\left(k^{1/10}\log
k\right)\right\\}=(k!)^{\ell}\cdot O\left(k^{-4/5}\right).$

Therefore

$\displaystyle\displaystyle\mathbb{E}[X^{2}]$
$\displaystyle\displaystyle=(k!)^{\ell}\sum_{b=0}^{k\ell}\left[\frac{\binom{k^{2}\ell-(2k\ell-b)}{m-(2k\ell-b)}}{\binom{k^{2}\ell}{m}}\sum_{\vec{b}\in
S_{b}}\prod_{c=1}^{\ell}\binom{k}{b_{c}}h(k-b_{c},k-b_{c})\right]$
$\displaystyle\displaystyle=(k!)^{\ell}\left[{\sum_{0\leq b\leq
k^{1/10}}\left(1+O(k^{-4/5})\right)(k!)^{\ell}p^{2k\ell-b}\exp\left\\{\ell\left(1-\frac{2}{p}\right)\right\\}\frac{\ell^{b}}{b!}+(k!)^{\ell}\cdot
O\left(k^{-4/5}\right)}\right]$
$\displaystyle\displaystyle=\left(1+O(k^{-4/5})\right)(k!)^{2\ell}p^{2k\ell}\exp\left\\{\ell\left(1-\frac{2}{p}\right)\right\\}\sum_{0\leq
b\leq k^{1/10}}p^{-b}\frac{\ell^{b}}{b!}$
$\displaystyle\displaystyle=\left(1+O(k^{-4/5})\right)(k!)^{2\ell}p^{2k\ell}\exp\left\\{\ell\left(1-\frac{2}{p}\right)\right\\}\cdot\left(\exp\left\\{\frac{\ell}{p}\right\\}+O(k^{-4/5})\right)$
$\displaystyle\displaystyle=\left(1+O(k^{-4/5})\right)(k!)^{2\ell}p^{2k\ell}\exp\left\\{\ell\left(1-\frac{1}{p}\right)\right\\}$
$\displaystyle\displaystyle\sim\mathbb{E}[X]^{2}$

For $\displaystyle\mathbb{E}[Y^{2}]$ we find an exact expression to be
cumbersome, but the following upper bound will suffice:

$\displaystyle\displaystyle\mathbb{E}[Y^{2}]$
$\displaystyle\displaystyle\leq\sum_{\begin{subarray}{c}0\leq i,j\leq k\\\
0\leq b\leq k\ell\\\ \vec{b}\in
S_{b}\end{subarray}}\frac{\binom{k^{2}\ell-(2k\ell-(i+j)\ell-b)}{m-(2k\ell-(i+j)\ell-b)}}{\binom{k^{2}\ell}{m}}\left(\frac{k!}{i!}\right)^{\ell}\prod_{c=1}^{\ell}\binom{k-i}{b_{c}}\binom{k-b_{c}}{j}h(k-j-
b_{c},k-i-b_{c}-2j)$ (2.8)

The term corresponding to a tuple $\displaystyle(i,j,b,\vec{b})$ above is an
upper bound on the contribution to $\displaystyle\mathbb{E}[Y^{2}]$ due to
pairs of permutations $\displaystyle(P,P^{\prime})$ such that $\displaystyle
P$ fixes $\displaystyle i$ vertices per part, $\displaystyle P^{\prime}$ fixes
$\displaystyle j$ vertices per part, and $\displaystyle P$ and $\displaystyle
P^{\prime}$ share a total of $\displaystyle b$ edges where $\displaystyle
b_{c}$ of the shared edges are between $\displaystyle V_{c}$ and part
$\displaystyle V_{c+1}$. The first factor is the edge probability, and the
next factor is the number of choices for $\displaystyle P$. The next factor is
an upper bound on the number of choices for $\displaystyle P^{\prime}$.
Indeed, we choose the edges of $\displaystyle P^{\prime}$ from $\displaystyle
V_{c}$ to $\displaystyle V_{c+1}$ by first choosing $\displaystyle b_{c}$
edges of $\displaystyle P$ to be shared, then we choose $\displaystyle j$
vertices in $\displaystyle V_{c}$ to be fixed by $\displaystyle P^{\prime}$,
and finally we choose a matching between the remaining vertices (the vertices
of $\displaystyle V_{c}\cup V_{c+1}$ that are not fixed by $\displaystyle
P^{\prime}$ and are not endpoints of the $\displaystyle b_{c}$ shared edges
already chosen). This matching must avoid any edges of $\displaystyle P$, and
the vertices to be matched induce at least $\displaystyle k-i-b_{c}-2j$ edges
of $\displaystyle P$, explaining the last factor above.

We will now estimate the significant terms in (2.8). Assume $\displaystyle
i,j,b\leq k^{1/10}$. Then by Fact 2.1

$\displaystyle\displaystyle\frac{\binom{k^{2}\ell-(2k\ell-(i+j)\ell-b)}{m-(2k\ell-(i+j)\ell-b)}}{\binom{k^{2}\ell}{m}}$
$\displaystyle\displaystyle=p^{2k\ell-(i+j)\ell-b}\exp\left\\{\frac{(2k\ell-(i+j)\ell-b)^{2}}{2k^{2}\ell}\left(1-\frac{1}{p}\right)+O\left(\frac{1}{k}\right)\right\\}$
$\displaystyle\displaystyle=p^{2k\ell-(i+j)\ell-b}\exp\left\\{2\ell\left(1-\frac{1}{p}\right)+O\left(k^{-9/10}\right)\right\\}.$

Next we estimate

$\displaystyle\displaystyle h(k-j-b_{c},k-i-
b_{c}-2j)=\left(1+O\left(k^{-4/5}\right)\right)\frac{(k-j-b_{c})!}{e}$

by Fact 2.2. So the product in (2.8) is

$\displaystyle\displaystyle\prod_{c=1}^{\ell}\binom{k-i}{b_{c}}\binom{k-b_{c}}{j}h(k-j-
b_{c},k-i-b_{c}-2j)$
$\displaystyle\displaystyle=\left(1+O\left(k^{-4/5}\right)\right)\prod_{c=1}^{\ell}\frac{(k-i)_{b_{c}}}{b_{c}!}\frac{(k-b_{c})_{j}}{j!}\frac{(k-j-
b_{c})!}{e}$
$\displaystyle\displaystyle\leq\left(1+O\left(k^{-4/5}\right)\right)\left(\frac{k!}{ej!}\right)^{\ell}\prod_{c=1}^{\ell}\frac{1}{b_{c}!}.$
(2.9)

The sum of terms in (2.8) corresponding to small $\displaystyle i,j,b$ is at
most

$\displaystyle\displaystyle\left(1+O\left(k^{-4/5}\right)\right)\sum_{\begin{subarray}{c}0\leq
i,j,b\leq k^{1/10}\\\ \vec{b}\in
S_{b}\end{subarray}}p^{2k\ell-(i+j)\ell-b}\exp\left\\{2\ell\left(1-\frac{1}{p}\right)\right\\}.\left(\frac{k!}{i!}\right)^{\ell}\left(\frac{k!}{ej!}\right)^{\ell}\prod_{c=1}^{\ell}\frac{1}{b_{c}!}$
$\displaystyle\displaystyle=\left(1+O\left(k^{-4/5}\right)\right)\exp\left\\{\ell\left(1-\frac{2}{p}\right)\right\\}\sum_{0\leq
i,j,b\leq
k^{1/10}}p^{2k\ell-(i+j)\ell-b}\left(\frac{k!}{i!}\right)^{\ell}\left(\frac{k!}{j!}\right)^{\ell}\frac{\ell^{b}}{b!}$
$\displaystyle\displaystyle\leq\left(1+O\left(k^{-4/5}\right)\right)(k!)^{2\ell}p^{2k\ell}\exp\left\\{\ell\left(1-\frac{1}{p}\right)\right\\}\sum_{0\leq
i,j,b}\frac{p^{-i\ell}}{(i!)^{\ell}}\cdot\frac{p^{-j\ell}}{(j!)^{\ell}}\cdot\frac{(\ell/p)^{b}}{b!}$
$\displaystyle\displaystyle=\left(1+O\left(k^{-4/5}\right)\right)(k!)^{2\ell}p^{2k\ell}\exp\left\\{\ell\left(1-\frac{1}{p}\right)\right\\}f_{\ell}\left(\frac{1}{p}\right)^{2}$
$\displaystyle\displaystyle\sim\mathbb{E}[Y]^{2}$

where on the second-to-last line we have used

$\displaystyle\sum_{0\leq
i}\frac{p^{-i\ell}}{(i!)^{\ell}}=f_{\ell}\left(\frac{1}{p}\right),\qquad\qquad\sum_{0\leq
b}\frac{(\ell/p)^{b}}{b!}=\exp\left\\{\frac{\ell}{p}\right\\}.$

It remains to show that the sum of all other terms (i.e. terms where
$\displaystyle i,j,$ or $\displaystyle b$ is at least $\displaystyle
k^{1/10}$) is negligible compared to $\displaystyle\mathbb{E}[Y]^{2}$, which
is of order $\displaystyle(k!)^{2\ell}p^{2k\ell}$. Note that by Fact 2.1

$\displaystyle\displaystyle\frac{\binom{k^{2}\ell-(2k\ell-(i+j)\ell-b)}{m-(2k\ell-(i+j)\ell-b)}}{\binom{k^{2}\ell}{m}}$
$\displaystyle\displaystyle=p^{2k\ell-(i+j)\ell-b}\exp\left\\{\frac{(2k\ell-(i+j)\ell-b)^{2}}{2k^{2}\ell}\left(1-\frac{1}{p}\right)+O\left(\frac{1}{k}\right)\right\\}$
$\displaystyle\displaystyle=O\left(p^{2k\ell-(i+j)\ell-b}\right).$

Thus, the sum (over $\displaystyle\vec{b}$) of terms corresponding to a fixed
triple $\displaystyle(i,j,b)$ in line (2.8) big-O of

$\displaystyle\displaystyle p^{2k\ell-(i+j)\ell-b}\sum_{\vec{b}\in
S_{b}}\left(\frac{k!}{i!}\right)^{\ell}\prod_{c=1}^{\ell}\binom{k-i}{b_{c}}\binom{k-b_{c}}{j}(k-j-
b_{c})!$ $\displaystyle\displaystyle\leq
p^{2k\ell-(i+j)\ell-b}\left(\frac{k!}{i!}\right)^{\ell}\left(\frac{k!}{j!}\right)^{\ell}\sum_{\vec{b}\in
S_{b}}\prod_{c=1}^{\ell}\frac{1}{b_{c}!}$
$\displaystyle\displaystyle=\left((k!)^{2\ell}p^{2k\ell}\right)\cdot\left(\frac{\ell^{b}}{p^{(i+j)\ell+b}(i!)^{\ell}(j!)^{\ell}b!}\right).$

It is easy to see that if $\displaystyle i,j$ or $\displaystyle b$ is at least
$\displaystyle k^{1/10}$ then the second factor above is
$\displaystyle\exp\left\\{-\Omega\left(k^{1/10}\log k\right)\right\\}$. Since
the number of triples $\displaystyle(i,j,b)$ is polynomial in $\displaystyle
k$, the sum of all such terms (i.e. where $\displaystyle i,j$ or
$\displaystyle b$ is at least $\displaystyle k^{1/10}$) is $\displaystyle
o\left((k!)^{2\ell}p^{2k\ell}\right)$ which is a negligible contribution to
$\displaystyle\mathbb{E}[Y^{2}]$. Therefore we have
$\displaystyle\mathbb{E}[Y^{2}]\sim\mathbb{E}[Y]^{2}.$

##  3 Remarks and Open Problems

The reader should note that we did not use a “binomial” random construction
(e.g. keep each edge of $\displaystyle D_{k,\ell}$ with probability
$\displaystyle p$ independently) because such a model lacks the concentration
we need here. Indeed, for example Janson ([3]) showed that the number of
perfect matchings in $\displaystyle G(n,p)$ is not concentrated even when it
is quite large, while the number of perfect matchings of $\displaystyle
G(n,m)$ is concentrated. We tried to use a binomial random construction and
found that the second moments were too large, which in light of Janson’s
result makes sense (for example derangements in our graph are just a union of
several perfect matchings on bipartite graphs).

There are still interesting open problems in [1]. In particular it is still
open whether $\displaystyle S$, the set of possible ratios
$\displaystyle(d/p)_{G}$, is equal to $\displaystyle\mathbb{Q}\cap[0,1/2]$.
Here we would like to pose another open problem that is mostly unrelated to
our result. In particular, we ask about stability for digraphs whose ratio
$\displaystyle(d/p)_{G}$ is close to $\displaystyle 1/2$: do such digraphs
have to resemble the blow-up of a directed graph?

## References

  * [1] M. Bucic, P. Devlin, M. Hendon, D. Horne, and B. Lund.  Perfect matchings and derangements on graphs, 2019.
  * [2] A. Frieze and M. Karoński.  Introduction to random graphs.  Cambridge University Press, Cambridge, 2016.
  * [3] S. Janson.  The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph.  Combin. Probab. Comput., 3(1):97–126, 1994.


Copyright 2022 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).

IberLEF 2022, September 2022, A Coruña, Spain.

[orcid=0000-0001-6900-1931<EMAIL_ADDRESS>] [1] [1]

<EMAIL_ADDRESS>orcid=0000-0002-0549-8771, ] [1]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

[1]Corresponding author. [1]These authors contributed equally.

# Deep Learning Brasil at ABSAPT 2022: Portuguese Transformer Ensemble
Approaches

Juliana Resplande Sant’Anna Gomes  Eduardo Augusto Santos Garcia  Adalberto
Ferreira Barbosa Junior  Ruan Chaves Rodrigues  Diogo Fernandes Costa Silva
Dyonnatan Ferreira Maia  Nádia Félix Felipe da Silva  Arlindo Rodrigues Galvão
Filho  Anderson da Silva Soares  Institute of Informatics, Federal University
of Goiás, Brazil  Faculty of Informatics, University of the Basque Country
(UPV/EHU), Spain

(2022)

###### Abstract

Aspect-based Sentiment Analysis (ABSA) is a task whose objective is to
classify the individual sentiment polarity of all entities, called aspects, in
a sentence. The task is composed of two subtasks: Aspect Term Extraction
(ATE), identify all aspect terms in a sentence; and Sentiment Orientation
Extraction (SOE), given a sentence and its aspect terms, the task is to
determine the sentiment polarity of each aspect term (positive, negative or
neutral). This article presents we present our participation in Aspect-Based
Sentiment Analysis in Portuguese (ABSAPT) 2022 at IberLEF 2022. We submitted
the best performing systems, achieving new state-of-the-art results on both
subtasks.

###### keywords:

Aspect-Based Sentiment Analysis Transformer architecture Portuguese Natural
Language Processing

##  1 Introduction

Aspect-based Sentiment Analysis (ABSA) can be defined as a fine-grained
approach to sentiment analysis. In this task, instead of attempting to
classify an entire sentence under a single sentiment polarity, we must
classify the individual sentiment polarity of all tokens that make a
significant contribution to the overall sentiment of the sentence. In the task
terminology, these tokens are called the aspect terms of a sentence.

Inspired on the format of SemEval-2014 Task 4 [1], the Aspect-Based Sentiment
Analysis in Portuguese (ABSAPT) 2022 at IberLEF 2022 features two subtasks.
The first subtask is called Aspect Term Extraction (ATE): given a set of
sentences, the task is to identify all aspect terms present in each sentence.
In this article, we present our participation at the Aspect-Based Sentiment
Analysis in Portuguese (ABSAPT) 2022 at IberLEF 2022. We submitted the best
performing systems, achieving new state-of-the-art results on all subtask.

The second subtask is called Sentiment Orientation Extraction (SOE): given a
set of sentences that have already been annotated for their aspect terms, the
task is to determine the sentiment polarity of each aspect term (positive,
negative or neutral). This subtask is also known as Aspect Term Polarity
Classification (ATP) [1, 2] or Aspect Sentiment Analysis (ASA) [3].

The remainder of this article is structured as follows: In the next section,
Related Work, we go into the previous research that supported our approach;
Next, under Dataset, we present a detailed analysis of the training data
provided for both subtasks. Moreover, under Methodology, Experimental setup
and Results, we articulate our strategy to address each subtask; Finally,
under Conclusion, we bring an overview of the results and a proposal for
future work.

##  2 Related Work

ATE ABSITA [4] was the EVALITA 2020 [5] shared task on Aspect Term Extraction
and Aspect-Based Sentiment Analysis. Both the first-ranked team [6] and the
second-ranked team [7] had the approach of simply framing Aspect Term
Extraction as a Named Entity Recognition task, and then fine-tuning state-of-
the-art Transformer models on the training data for the task. We followed a
similar approach during our participation at ABSAPT 2022.

ATE ABSITA [4] also features a subtask similar to Sentiment Orientation
Extraction (SOE) at ABSAPT 2022. The first-ranked team [6] framed it as a
problem of text classification, under the premise that the portion of the text
that surrounds each aspect should have the same overall sentiment as the
aspect itself.

Although we also experimented with this approach, our best performing system
at the Sentiment Orientation Extraction subtask framed it as a text generation
problem, similar to what was done by Zhang et al. [8] and Chebolu et al. [9].

##  3 Dataset

The dataset was taken from TripAdvisor reviews, specifically from the
hospitality industry, consisting of hotel and room reviews. Table 1 is an
example of a positive polarity from the train dataset. The structure of the
data is as follows: id, review, polarity, aspect, start position and end
position.

Table 1: Sentiment Orientation Extraction example. Start and End positions are abbreviated into start pos. and end pos. id |  review | polarity | aspect | start pos. | end pos.
---|---|---|---|---|---
2414 |  Hospedei-me em maio nesse hotel
pela terceira vez … | 1 | hotel | 26 | 31

For the Aspect Term Extraction task, we build a NER dataset converting the
original train data to the BIO/IOB format (Inside, Outside, Beginning), a
common tagging format where each token can be classified with the prefixes _B_
, _I_ and _O_. The _B_ prefix indicates the begging of a new classification
chunk, the _I_ prefix indicates that the token is inside a previous chunk and
the _O_ tag indicates that the token doesn’t belong to any class or chunk. An
example of the annotation is shown in Table 2.

Table 2: Example of an Aspect Term Extraction sentence converted to the BIO
tagging format.

A | estrutura | do | hotel | é | muito | boa. | A | piscina | é | excelente | e | os | quartos | também.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
O | O | O | B-ASPECT | O | O | O | O | B-ASPECT | O | O | O | O | B-ASPECT | O

Analyzing the provided dataset in search of imbalanced data that could be
exploited, we found some noteworthy cases. Polarity, for example, is
unbalanced towards 1 (positive), representing about 68% of the dataset.
Furthermore, the distributed polarity with respect to the aspects is also
unbalanced, as seen in Figure 1.

Figure 1: Aspect polarity heat map from top 15 ocorrences.

Another interesting point is the aspect term occurrences. Although the dataset
contains 77 unique aspects, the top 15 represent 79% of the data, as shown in
Figure 2. Furthermore, the aspect term position is not evenly distributed
across the dataset, as its occurrence is more common at the beginning of a
review.

Figure 2: The number of occurrences of the 15 most common aspect occurrences
in the training dataset.

Concerning the Aspect Term Extraction, we found that the distribution of
aspects in the reviews has a mean of 3.7 and a standard deviation of 1.7, As
seen in Figure 3. In contrast, the distribution of words has a mean of 68 and
a standard distribution of 21. The low amount of aspects per review can
provide an extra challenge for a model to learn accurately, since most of the
words in a given review will not be an aspect.

Figure 3: The frequency distribution of the total number of aspects for each
unique sentence on the training dataset.

##  4 Methodology

We adopt different methodologies in the ATE and SOE tasks. For ATE, we treat
it primarily as a single sentence tagging task [10]. For SOE, we test two
distinct strategies: treating SOE as sentence pair classification or as
conditional text generation.

###  4.1 Task 1 - Aspect Term Extraction

For Aspect Term Extraction, we assess four training considerations: training
strategy, Transformer model, dataset preprocessing and dataset configuration,
in which we explain in the next subsections.

####  4.1.1 Training strategy

For the objective of Aspect Term Extraction, we transform the dataset to
behave as a Named Entity Recognition task (NER) [6], where we classify chunks
of the sentence with the objective to identify which group of tokens
represents an aspect. The reviews are tokenized and each aspect is tagged with
the BIO format, we train a transformer model using the Hugging Face
transformers library [11].

####  4.1.2 Transformer model

We consider evaluating the following Tranformer-based models with Portuguese
support: Bertimbau base ; mDeBERTa v3 base; and our own RoBERTa base in
Portuguese trained on BrWaC and the Portuguese portion of OSCAR corpora. Also
an mDeBERTA pretrained model on Evalita, MAMs and Semeval datasets, which can
be considered as mDEBERTA trained previously on additional data.

####  4.1.3 Dataset preprocessing

The original competition dataset is separated by aspect, for the conversion to
a NER dataset we join all the aspects of reviews with the same text and each
unique review it’s used as a single training example. We use the NTLK library
[12] to tokenize and generate the BIO annotated dataset for a model’s
training.

####  4.1.4 Dataset configuration

In addition to the dataset in Portuguese of the ABSAPT 2022 we used external
ABSA datasets such as Evalita, MAMs, Semeval 2014, 2015, and 2016 tasks.

###  4.2 Task 2 - Sentiment Orientation Extraction

For Sentiment Orientation Extraction, we assess four training considerations:
training strategy, review preprocessing, Transformer model and dataset
configuration, in which we explain in the next subsections.

####  4.2.1 Training strategy

SOE can be targeted with masked language models or autoregressive language
models [8, 9].

If SOE is targeted with a masked language model, following Devin et. al. [10],
the review and the aspect term are concatenated with a sentence separator
token ( [SEP] ), and then the resulting sequence of tokens is assigned a
sentiment polarity.

However, if SOE is targeted with an autoregressive language model, it will
generate sentiment polarity labels starting from a prompt in the format of
“Review: [review content] Aspect: [aspect term] Polarity:”, which should be
completed with either “positive”, “negative” or “neutral”. This is done either
in a zero-shot fashion, in the case of GPT-3, or after fine-tuning a sequence-
to-sequence Transformer model to examples given in this format, such as PTT5
[13].

####  4.2.2 Review preprocessing

We tested two review preprocessing setups for the SOE task. In the first one,
the entire review is concatenated with the aspect term, just as described
under our training strategy. In the second one, only the portion of the review
that is relevant to the aspect term is concatenated with it. Following Rosa et
al. [6], we implement this by concatenating with the aspect term only the
sentence in which the aspect term is located.

####  4.2.3 Transformer model

In order to determine the Transformer model to be used, we consider Tranformer
models with Portuguese support: Bertimbau base; Bertimbau large; PPT5 base;
PTT5 large; mDeBERTa base; XLM-RoBERTa base; LaBSE, Canine-c, RemBERT.

####  4.2.4 Dataset configuration

We test whether to use external data using B2W dataset and ABSA datasets, such
as Evalita, MAMs, Semeval 2014, 2015, and 2016 tasks, by extending pseudo-
training subsets on 5.2.

We additionally tested mDeBERTA pretrained model on MNLI and XNLI provided by
MoritzLaurer/mDeBERTa-v3-base-mnli-xnli, which can be considered as mDEBERTA
trained previously on additional data: MNLI and XNLI.

Moreover, we attempt to perform data augmentation over the original task
dataset throuh target swap, according to what has been suggested by Liesting
et al. [14]. This approach consists of artificially increasing the amount of
sentences on the dataset by replacing aspect terms by others belonging to the
same category. As there are no annotated categories to the reviews on the
ABSAPT 2022 dataset, we tried to approximate such categories through topic
modeling.

##  5 Experimental setup

In experiments, two V100 GPUs (32 GB) were employed, one for each task.

###  5.1 Task 1 - Aspect Term Extraction

To evaluate the quality of the models we created a pseudo test-training splits
of 30/70 of training set where we make sure that each split has unique
sentences, this avoids leakage of data between the pseudo-train and pseudo-
test subsets. Including the external data, we create two subsets for the
training of the ATE task:

  * •

Portuguese subset: 70% random split of the original training-set of the
competition dataset.

  * •

multilingual subset: All external data from Evalita, MAMs, Semeval, plus the
Portuguese subset.

All models are fine-tuned in using the HuggingFace Transformer library with a
batch-size of 8, learning-rate of 3e-5 (BERTimbau, RoBERTa) or 4e-5 (mDeBERTa)
for 8 epochs. The evaluation occurred in the 30% random split of the original
training-set of the competition dataset.

###  5.2 Task 2 - Sentiment Orientation Extraction

We evaluate the approaches mentioned under subsection 4.2 by splitting the
training set for the shared task into a new training set and validation set
for evaluation purposes. We experiment with three different ways of splitting
it:

  * •

subset 1: Random approach. The training set for the shared task is arbitrarily
split into a new training and validation set.

  * •

subset 2: We are careful to keep the same proportion of reviews of each
polarity on each split.

  * •

subset 3: Besides polarity, we also try to keep the same ratio of aspect terms
between the splits.

On all subsets, we attribute 70% of the reviews to the new training set, and
30% of the reviews to the validation set.

As shown in section 6, the best results for the SOE task in our experiments
were achieved by PTT5 Large with the conditional text generation training
strategy, while taking the entire review and the aspect term as an input to
the model, and without using any external data.

We take this model and fine-tune it under four distinct combinations of
learning rate and random seed: $\\{3e-4,7\\}$, $\\{1e-4,5\\}$ and
$\\{5e-5,8\\}$. Afterward, we produce the final submission through a majority
voting ensemble of the predictions of the four fine-tuned models.

Ensemble training code is available on https://github.com/ju-
resplande/dlb_absapt2022.

##  6 Results

###  6.1 Task 1 - Aspect Term Extraction

We evaluate the training strategies in 4.1, in terms of the following metrics:
accuracy (acc.), precision-macro (precision), recall-macro (recall) and
f1-macro (f1) for the pseudo-test split of the competition dataset.

The results of the internal evaluation in the pseudo-test splits of each
combination of model and dataset tested can be found in Table 3.

Table 3: Best results on the ATE task. The symbol {MAMs, Evalita, Semeval}∗ refers to mDeBERTa previously trained on external datasets, explained in 4.1.2. model | external data | acc. | precision | recall | f1
---|---|---|---|---|---
BERTimbau base | - | 98.2 | 78.1 | 87.8 | 82.6
RoBERTa PT base | - | 98.4 | 80.8 | 90.7 | 85.5
mDeBERTa base | MAMs, Evalita, Semeval | 98.4 | 79.1 | 94.0 | 85.9
mDeBERTa base | {MAMs, Evalita, Semeval}∗ | 98.5 | 81.4 | 90.1 | 85.5

The final submission was created using an ensemble of the 3 best models, using
a simple median of the label probabilities output by the models for each token
in a review, the competition results for the ATE task can be found in table 4.

Table 4: Competition final results for the Task 1 (ATE). team_name | acc
---|---
TeamDeepLearningBrasil | 67.1448
Teampiln | 65.4974
TeamUFSCAR | 59.3715
TeamPeAm | 33.8243
TeamMachadoPardo | 22.1050
TeamUFPR | 17.1908
TeamOwl | 2.6265

###  6.2 Task 2 - Sentiment Orientation Extraction

We evaluate the training strategies in 4.2, according to subsets 5.2, in terms
of the following metrics: accuracy (acc.), f1-macro (f1), and f1 on each
class; positive - f1(pos), neutral - f1(neu), and negative - f1(neg).

Table 5 illustrates 3 best results for each training subset described under
subsection 5.2, and also has zero-shot results for GPT-3 [15], which has not
been fine-tuned on any of our data. The best results for each specific metric
are highlighted in bold.

Table 5: Best results on training subsets. We also include zero-shot results obtained with GPT-3 [15] . The symbol {MNLI, XNLI}∗ refers to mDeBERTa previously trained on MNLI and XNLI, explained in 4.2.4. subset | model | external data | acc. | f1 | f1(pos) | f1(neu) | f1(neg)
---|---|---|---|---|---|---|---
subset 1 | PTT5 large | - | 86.8 | 78.8 | 92.6 | 62 | 81.7
PTT5 large | target swap | 86.3 | 78.3 | 92.4 | 63 | 79.6
PTT5 base | target swap | 86.3 | 78.2 | 92.6 | 63.1 | 78.8
GPT-3 | - | 80.0 | 66.0 | 90.0 | 36.0 | 73.0
subset 2 | mDeBERTa base | {MNLI, XNLI}∗ | 85.1 | 75.9 | 92.4 | 57.1 | 78.2
mDeBERTa base | MAMs, Evalita, Semeval | 84.6 | 76.6 | 91.6 | 55.2 | 83
mDeBERTa base | - | 83.3 | 73.5 | 91.2 | 52.1 | 77.1
subset 3 | PTT5 large | - | 77.4 | 75.6 | 82.3 | 60.6 | 83.8
mDeBERTa base | MAMs, Evalita, Semeval | 76.9 | 75.6 | 82 | 62 | 82.8
mDeBERTa base | {MNLI, XNLI}∗ | 74.2 | 72.1 | 81.6 | 58.1 | 76.7

####  6.2.1 GPT-3

It is important to mention that we did not translate the reviews into English
before turning them into prompts for GPT-3 through the OpenAI’s text
completion API endpoint 111https://beta.openai.com/docs/api-
reference/completions. The prompts were designed as described on subsection
4.2. Furthermore, due to time constraints, we did not perform predictions for
the entire test set, but only for a subset of 391 reviews.

####  6.2.2 Review preprocessing

For all the best models tested, we achieved the best results by taking the
entire review and the aspect term as an input, instead of trying to extract
from the review only the sentences which are relevant to the aspect term.

####  6.2.3 PTT5 Large

On subset 1 and subset 3, PTT5 large without external data provides the best
results, and mDeBERTa base surpasses PTT5 large only on subset 2. Therefore,
in our final submission, we opted for a conditional text generation approach
where we build a voting ensemble of PTT5 large models, as described under
subsection 5.2.

Table 6 shows the final results of the competition for the SOE task and the
metrics of our final submission in comparison with other teams.

Table 6: Competition final results for the Task 2 (SOE). team_name | bacc | f1 | precison | recall
---|---|---|---|---
TeamDeepLearningBrasil | 82.3756 | 81.7988 | 81.3144 | 82.3756
Teampiln | 78.8619 | 77.4794 | 76.5911 | 78.8619
TeamUFSCAR | 62.8992 | 61.2248 | 65.5697 | 62.8992
TeamPeAm | 62.8992 | 61.2248 | 65.5697 | 62.8992
TeamUFPR | 62.8992 | 61.2248 | 65.5697 | 62.8992
TeamOwl | 53.5995 | 57.2803 | 68.9396 | 53.5996

##  7 Conclusion

We submitted the best performing system on ABSAPT 2022 at IberLEF 2022,
achieving new state-of-the-art results on both ATE and SOE tasks. For ATE, we
used an ensemble of RoBERTa and mDeBERTa’s models trained in Portuguese and
multilingual datasets, respectively. For SOE, we employed a voting ensemble of
PTT5 large without external data.

In future work, we plan to experiment with a coreference resolution step to
improve aspect term extraction by removing ambiguity from the reviews [16]. It
may also be interesting to consider fine-tuning Transformer models for both
subtasks simultaneously through a multi-task learning framework [17].

###### Acknowledgements.

This work has been supported by the AI Center of Excellence (Centro de
Excelência em Inteligência Artificial – CEIA) of the Institute of Informatics
at the Federal University of Goiás (INF-UFG).

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2022 111 01 June

Net-baryon fluctuations with cumulants up to third order in Pb–Pb collisions

ALICE CollaborationALICE Collaboration

Fluctuation measurements are important sources of information on the mechanism
of particle production at LHC energies. This article reports the first
experimental results on third-order cumulants of the net-proton distributions
in Pb–Pb collisions at a center-of-mass energy $\sqrt{s_{\mathrm{NN}}}=5.02$
TeV recorded by the ALICE detector. The results on the second-order cumulants
of net-proton distributions at $\sqrt{s_{\mathrm{NN}}}=2.76$ and $5.02$ TeV
are also discussed in view of effects due to the global and local baryon
number conservation. The results demonstrate the presence of long-range
rapidity correlations between protons and antiprotons. Such correlations
originate from the early phase of the collision. The experimental results are
compared with HIJING and EPOS model calculations, and the dependence of the
fluctuation measurements on the phase-space coverage is examined in the
context of lattice quantum chromodynamics (LQCD) and hadron resonance gas
(HRG) model estimations. The measured third-order cumulants are consistent
with zero within experimental uncertainties of about 4% and are described well
by LQCD and HRG predictions.

##  1 Introduction

Predictions based on the theory of the strong interaction, QCD, imply that, at
sufficiently high energy densities, nuclear matter transforms into a state
called quark–gluon plasma (QGP) [1, 2], where chiral symmetry is restored and
quarks and gluons are deconfined. Ultrarelativistic heavy-ion collisions are
ideal environments to study the phase diagram of strongly interacting matter
and the physics of the QGP state as a function of temperature ($T$) and baryon
chemical potential ($\mu_{\rm B}$). While the QCD phase diagram is largely
unknown for $\mu_{\rm B}>450$ MeV, the region below that value down to
$\mu_{\rm B}=0$ has been well explored theoretically [2, 3] and experimentally
[4]. In that region, the chiral phase transition is most likely a continuous
crossover with pseudo-critical temperature $T_{\mathrm{pc}}\approx 157$ MeV at
$\mu_{\rm B}=0$ [5, 6]. Near $\mu_{\rm B}=0$ the properties of the QCD phase
transition depend on the number of quark flavors and their masses. For
vanishing masses of the light quarks (u,d) the transition is of second order
and belongs to the universality class of three-dimensional O(4)-symmetric spin
models [7]. On the other hand, for the small physical quark masses this
transition turns into a smooth crossover. To date, there is no experimental
confirmation of the crossover nature of the transition. Moreover, the
smallness of the physical quark masses may leave traces of critical behavior
also for a continuous transition. Therefore, a significant effort at the RHIC
and LHC colliders is concentrated on quantifying the nature of the phase
transition in the small $\mu_{\rm B}$ region.

For large $\mu_{\rm B}$ the phase diagram may exhibit a “critical endpoint
(CEP)”. The search for the CEP is one of the main physics goals of the beam
energy scan programs at RHIC [8, 9], at the CERN SPS [10], and at FAIR [11].

Recent LQCD [12, 13, 14] calculations of chiral susceptibilities for small
quark masses exhibit a clear peak at $T_{\mathrm{pc}}$, consistent with the
chemical freeze-out temperature extracted by the analysis of hadron
multiplicities [4, 15] measured in central Pb–Pb collisions by the ALICE
experiment. This suggests that the chemical freeze-out occurs very close to
the chiral phase transition at LHC energies. Critical signals associated with
this phenomenon can be linked with long-range correlations and increasing
multiplicity fluctuations due to the existence of the massless modes of the
second-order phase transition [16, 17]. In particular, the fluctuations of the
conserved charges are very sensitive probes for the equation of state and can
be directly related to the thermodynamic susceptibilities, which are
calculable in the framework of LQCD. The quark-number susceptibilities are
defined as the derivatives of the reduced QCD pressure with respect to the
reduced chemical potentials ($\hat{\mu}=\mu/T$) of the conserved charges

$\chi_{klmn}^{\mathrm{B,S,Q,C}}=\frac{\partial^{(k+l+m+n)}(P(\hat{\mu}_{\rm
B},\hat{\mu}_{\rm S},\hat{\mu}_{\rm Q},\hat{\mu}_{\rm
C})/T^{4})}{\partial\hat{\mu}_{\rm B}^{k}\partial\hat{\mu}_{\rm
S}^{l}\partial\hat{\mu}_{\rm Q}^{m}\partial\hat{\mu}_{\rm
C}^{n}}\Big{|}_{\vec{\mu}=0}.$ (1)

Here the relevant conserved charges, represented by the chemical potentials,
are the electric charge Q, the baryon number B, the strangeness S, and the
charm C. These susceptibilities are studied experimentally in terms of the
ratios of the cumulants111The cumulants, $\kappa_{n}$, of net-baryon number,
$\Delta N_{\rm B}=N_{\rm B}-N_{\rm\overline{B}}$, are defined as the
coefficients in the Maclaurin series of the logarithm of the characteristic
function of $\Delta N_{\rm B}$ [18]. of net-charge distributions [19]. Here
and in the following, net-charge distribution stands for the difference
between the distributions of positively and negatively charged particles,
where “charge” refers to any additive quantum number. Signs of criticality due
to the proximity of the chiral crossover transition to a second-order
transition are expected to show up starting only with the sixth-order
cumulants of net-charge distributions [20, 16]. Measuring conserved charge
fluctuations in ultrarelativistic nuclear collisions is a challenging task,
see Ref. [8] for a recent review. In addition to possible critical
fluctuations due to the proximity of a CEP or O(N) criticality, there are
several other dynamical signals, such as correlations due to baryon number
conservation [21, 22], volume fluctuations [18], thermal blurring [23],
resonance decays, which could overshadow the dynamical fluctuations of
interest.

The article is organized as follows. In Section 2, details are given about the
ALICE detector, the data set and the analysis procedure, such as event and
track selection criteria, particle identification and the efficiency
correction technique. In Section 3, results on second-order cumulants of net-
pion and net-kaon distributions are presented to demonstrate the influence of
resonance decays on the measured cumulants. The main results are contained in
the second- and third-order cumulants of net-proton distributions and compared
to theoretical expectations. The article concludes, in Section 4, with a
discussion on how the present results fit into the general strategy to get
information on possible critical behaviour near the QCD phase boundary and
with an outline of the next steps.

##  2 Experimental setup and data analysis

The measurements presented in this article are based on about 13 and 78
million minimum bias Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ and
$5.02$ TeV recorded with the ALICE detector [24, 25] in the years 2010 and
2015, respectively. Note that the running conditions have changed
significantly in 2015. The details presented in this article refer only to the
analysis of 2015 data (see Refs. [26, 27] for 2010). The minimum-bias trigger
condition is defined by requiring a coincidence of hits in both V0 detectors
[28] located on either side of the nominal interaction point along the beam
direction and covering the pseudorapidity intervals $2.8<\eta<5.1$ and
$-3.7<\eta<-1.7$. The definition of the centrality [29] is based on the signal
amplitudes measured in the V0 detectors, which are related to the collision
geometry and the number of participating nucleons through a Monte Carlo (MC)
simulation based on a Glauber model [30, 31]. Beyond event characterization,
the main sub-detectors used in the analysis are the Time Projection Chamber
(TPC) [32] for tracking and particle identification (PID), using the specific
energy loss $\textrm{d}E/\textrm{d}x$, and the Inner Tracking System (ITS)
[33, 34] for tracking and vertex determination.

Event and track selection criteria were applied to ensure optimal PID
performance via $\textrm{d}E/\textrm{d}x$ measurement and momentum ($p$)
resolution, as well as good track quality. The V0 [28] and the Zero Degree
Calorimeter [35] timing information was used to reject background due to
beam–gas interactions and parasitic beam–beam interactions. Along the beam
direction, events occurring within $\pm 0.15$ cm, where tracking is affected
by the central membrane of the TPC, and outside $\pm 7$ cm of the nominal
interaction point were discarded to keep detection efficiency uniform as a
function of event vertex position. The analysis uses tracks of charged
particles reconstructed using the ITS and the TPC in the pseudorapidity range
of $|\eta|<0.8$ and with full azimuthal acceptance. In 2015, the TPC
performance was affected by local space-charge distortions caused by the
accumulation of space charge originating from the gaps between adjacent
readout chambers [36]. Although these distortions are corrected on average
during reconstruction, their fluctuations caused a significant mismatch
between the TPC and ITS tracks. They affected all events detected within a
time interval of about 0.5 seconds after a collision, corresponding to the
full drift time of ions from the amplification region in the TPC readout
chambers to the central electrode. Collisions within such time intervals were
discarded using the event interaction time (timestamp) when also the mean
matching efficiency between the TPC and ITS for all tracks in an event was
less than 88% (the nominal TPC–ITS track matching efficiency is peaked at
90.5% with a small pseudorapidity and momentum dependence of a few percent).
By applying this condition about 7% of all events are rejected. Tracks were
accepted if they crossed at least 80 padrows out of a maximum of 159 and their
$\chi^{2}$ value per space point from the track fit is less than 3.5. To
improve the $\textrm{d}E/\textrm{d}x$ resolution and reduce the contributions
from track splitting222Track splitting occurs when a track is reconstructed as
two separate tracks due to poor $\textrm{d}E/\textrm{d}x$ resolution., two
additional track selection criteria were applied. The number of clusters used
for the $\textrm{d}E/\textrm{d}x$ calculation333Clusters that are very close
to the TPC readout chamber boundaries [36] or are from overlapping tracks are
not considered in the calculation of $\textrm{d}E/\textrm{d}x$. Therefore, the
number of clusters used to calculate $\textrm{d}E/\textrm{d}x$ may differ from
the number of clusters used for track selection. This quantity is particularly
important for the $\textrm{d}E/\textrm{d}x$ resolution. was set to more than
70. These criteria address both tracks crossing chamber boundaries and split
tracks, both leading to a deterioration of the $\textrm{d}E/\textrm{d}x$
measurement. To suppress contributions from secondary particles from weak
decays, the distance of closest approach (DCA) of the extrapolated track to
the primary vertex position was required to be less than 1 cm along the beam
direction, while in the transverse plane a transverse momentum ($p_{\rm T}$)
dependent DCA selection of less than (0.018 $\rm cm$ $+$ 0.035$p_{\rm
T}^{-1.01}$) with $p_{\rm T}$ in GeV/c was applied to account for the $p_{\rm
T}$ dependence of the DCA resolution [37]. In addition, daughter tracks from
reconstructed secondary weak-decay kink topologies were discarded.

In 2015, the ALICE TPC was operated at interaction rates up to 8 kHz during
Pb–Pb collisions. These high interaction rates resulted in a possible pile-up
of several interactions during the 90 $\mu$s drift time. The fraction of pile-
up events, which occur within a few cm of the primary vertex, the so-called
in-bunch pile-up, is less than 1‰ of the entire data set. Therefore, in-bunch
pile-up has a negligible impact on the results. Moreover, in-bunch pile-up
events were further suppressed thanks to the high-precision vertexing
capabilities of the ITS. Even slightly displaced multiple vertices originating
from interactions within the same bunch crossing can be discriminated. The
much more frequent out-of-bunch pile-up events occur anytime during the drift
time of the TPC, i.e., they are distributed equally along the time direction
within the TPC and cannot be suppressed by reconstructed secondary vertices in
the ITS. They lead to a significant deterioration in the measurements of the
$\textrm{d}E/\textrm{d}x$ of the triggered event caused by the baseline
fluctuations in the TPC [36]. These pile-up events affect about 20% of the
data collected in 2015 and are negligible for 2010. The resulting bias in the
measured $\textrm{d}E/\textrm{d}x$ is corrected to restore optimal PID
performance and to avoid the need to discard events with pile-up [36].

Protons are identified by their $\textrm{d}E/\textrm{d}x$ in the TPC and its
known momentum dependence. To overcome the misidentification problem caused by
overlapping $\textrm{d}E/\textrm{d}x$ distributions for different particles,
for instance at the crossings of kaons (K) and protons (p) or pions ($\pi$)
and protons, a novel experimental technique, the Identity Method (IM) [38, 39,
40] was used. With this method, weights, which are obtained from the fits of
$\textrm{d}E/\textrm{d}x$ distributions, are assigned to each track reflecting
the probability of a particle having a specific identity. Thereby all tracks
can be kept without applying any selection on the PID variable and no second
detector needs to be employed to identify the (anti)protons. In the present
analysis, particles in the momentum range 0.6–2.0 GeV/c were retained. With
particle identification based solely on the $\textrm{d}E/\textrm{d}x$
measurements in the TPC, the efficiencies are independent of momentum in the
selected range, and as high as about 91% for protons and about 83% for
antiprotons. The difference of 8% is due to the absorption of antiprotons in
the detector material. Both proton and antiproton detection efficiencies are
nearly independent of collision centrality and are uniform within the
kinematic acceptance used in this analysis. The cumulants of the net-proton
distribution are reconstructed using the IM as well. Further details on the
application of the IM to ALICE data are discussed in Ref. [27]. In Ref. [26],
the method used here was applied to the analysis of second-order cumulants of
net protons in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV. The
current analysis closely follows what is described there. The efficiency
correction for the cumulants is performed by using proton and antiproton
efficiencies in analytic formulas derived in Refs. [41, 42, 43] assuming
efficiency losses governed by the binomial statistics (binomial efficiency
loss). In order to ensure that this assumption is fulfilled, the ALICE
detector response [25] was studied in detail with a full MC simulation using
the HIJING event generator and the GEANT4 [44] transport software. The
response of the TPC detector for the kinematic range used in this analysis is
illustrated in the left panel of Fig. 1 as a correlation between the
reconstructed ($N_{\rm p}^{\rm rec}$) and the generated ($N_{\rm p}^{\rm
gen}$) number of protons, where the reconstructed protons are those detected
in the active area of the detector, as well as satisfying the event and track
quality criteria. The event and track selection plays a significant role in
the shape of the TPC detector response. For example, as discussed above, they
can reduce the contributions from track splitting, which causes correlated
$\textrm{d}E/\textrm{d}x$ measurements, and thus a deviation from a binomial
detector response.

Figure 1: (Left) Correlation between the reconstructed ($N_{\rm p}^{\rm rec}$)
and the generated ($N_{\rm p}^{\rm gen}$) number of protons for the most
central Pb–Pb collisions simulated using the HIJING model [45]. (Right)
Distribution of reconstructed proton number for a fixed value of $N_{\rm
p}^{\rm gen}=36$, where the fit demonstrates the deviation from a binomial
efficiency loss.

After detailed study and optimization of the event and track selection, only a
slight deviation from the binomial loss is observed, as shown in the right
panel of Fig. 1. A MC verification test was performed to estimate the impact
of this deviation in the final results on the net-proton cumulant
measurements. In this MC closure test, particles are generated, including
certain correlations such as the effect of baryon number conservation, and
reconstructed after they have passed through the detector simulated with
GEANT4. Then the efficiency correction is applied, and the generated and
corrected observables are compared. The comparison is shown in Fig. 2 for the
second- and third-order cumulant ratios of the net-proton distribution. The
efficiency-corrected results obtained from the MC reconstructed data are in
agreement with the results obtained from the MC generated data.

Figure 2: HIJING model [45] based calculations of the normalized second-order
cumulants of net protons as a function of pseudorapidity window ($\Delta\eta$)
(left) and ratio of third- to second-order cumulants (right) of net protons as
a function of collision centrality at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. The
results at the generated and reconstructed level are shown by the green closed
and open circles, respectively. The error bars represent statistical
uncertainties. The results after efficiency correction assuming binomial
efficiency losses [41, 42, 43] are shown by black open squares.

The statistical uncertainties assigned to the reconstructed cumulants were
determined using the sub-sample method as described in Ref. [27]. The fits to
the measured $\textrm{d}E/\textrm{d}x$ distributions, which are the only
inputs to the IM, are the dominant source of systematic uncertainty in the
ratios of the cumulants, for both second and third order. The observed maximum
deviation between fit variations [27] is 0.6% and 0.8% for the normalized
second-order cumulants within the momentum intervals of 0.6–1.5 GeV/c and
0.6–2.0 GeV/c, respectively, and 4% for the ratio of third- to second-order
cumulants in the momentum interval 0.6–1.5 GeV/c. The impact of possible
imperfections in the $\textrm{d}E/\textrm{d}x$ correction procedure mentioned
above is also included in this systematic uncertainty estimate by analyzing
the data retaining different fractions of events containing pile-up. The
uncertainties associated with the detection efficiencies of the (anti)protons
are also investigated by varying the detection efficiencies by an amount of
$\pm 2$% for protons and antiprotons separately. The resulting systematic
variation is less than 0.2% and 1.5% for the second- and third-order
cumulants, respectively. Other sources of systematic uncertainty are estimated
by varying the event and track selection criteria, resulting in a maximum
uncertainty of less than 1%. The final total systematic uncertainty is
obtained by adding in quadrature the individual maximum systematic deviations
from these three groups of independent contributions. For the third-order
cumulants, it varies between less than 0.5% for the most peripheral collisions
and a maximum of 3% for the most central collisions for the pseudorapidity
interval of $\Delta\eta=1.6$.

##  3 Results

As potential candidates for conservation of electric charge and strangeness,
results are reported for the pseudorapidity interval dependence of the second-
order cumulants of net-pions and net-kaons produced in central Pb–Pb
collisions.

Figure 3: Pseudorapidity interval dependence of the second-order cumulants of
net-pions (left) and net-kaons (right) normalized to the means (see text). The
ALICE data are shown as solid black circles while the blue solid and dashed
lines indicate the results from HIJING [45] model calculations with and
without resonance contributions, respectively. The error bars represent
statistical uncertainties and the boxes around the data points represent the
total systematic uncertainties.

The observations in these channels are quite striking because they shed light
on resonance decay contributions to fluctuations in Pb–Pb collisions at the
LHC. Figure 3 shows the pseudorapidity interval dependence of the normalized
second-order cumulants of net-pions and net-kaons compared with the results
from HIJING [45] with and without resonance contributions. A significant
effect of resonances, e.g., $\rho\rightarrow\pi^{+}\pi^{-}$ and
$\phi\rightarrow\rm K^{+}\rm K^{-}$, is clearly visible in both cases. In
fact, the decay of resonances into oppositely charged pion or kaon pairs
drastically reduces the fluctuations and dominates the second-order cumulants
of the respective net distributions. Therefore, to study the genuine electric
charge and strangeness fluctuations, first a quantitative understanding of the
resonance contributions is essential. On the other hand, there are no
resonances that decay into $\rm p\overline{\rm p}$ with a sizeable branching
ratio, therefore net-proton fluctuations are not obscured by this effect. It
has been argued in the literature [46] that net-proton fluctuations are good
proxies for net-baryon fluctuations, in particular for
$\sqrt{s_{\mathrm{NN}}}>$10 GeV. Also, total electric-charge conservation is
expected to have a negligible impact on the net-proton fluctuation
measurements, since the electric charge is mostly carried by the charged
pions, which are the most abundant species at LHC energies. The statistically
independent Poisson limit for net-baryon distributions is the Skellam
distribution, which is defined as the probability distribution of the
difference of two random variables, each generated from statistically
independent Poisson distributions [47, 48]. For net protons, the $n$th-order
cumulants of the Skellam distribution are given by

$\kappa^{\mathrm{Skellam}}_{n}(\mathrm{p}-\mathrm{\overline{p}})=\langle\mathrm{p}\rangle+(-1)^{n}\langle\mathrm{\overline{p}}\rangle,$
(2)

where $\langle\mathrm{p}\rangle$ and $\langle\mathrm{\overline{p}}\rangle$ are
the mean values of the proton and antiproton multiplicity distributions,
respectively. That means that even-order cumulants of the Skellam distribution
of the net protons are just the sum of the mean numbers of protons and
antiprotons. At LHC energies, these numbers are equal within 1% [49], and
therefore the normalized cumulants of the Skellam distribution with respect to
its second-order cumulant are zero for odd cumulants and unity for even
cumulants. At $T_{\mathrm{pc}}$ [5, 6], both the predictions based on LQCD and
the HRG [4] model agree with the Skellam baseline up to the third-order
cumulants of the net protons, reflecting independent Poissonian fluctuations.
The LQCD prediction [50], including the effect of dynamical quarks, shows a
significant deviation from the Skellam baseline for the fourth- and higher-
order cumulants, while the standard HRG does not contain such effects and
deviations from the Skellam baseline are only due to baryon number
conservation [51]. Fluctuations of conserved charges are meaningful only
within a limited phase space. They vanish in the full phase space, in order to
obey the conservation laws, and asymptotically approach the Poisson limit for
very small acceptance, where dynamical correlations are suppressed [41].
Therefore, the fluctuations of net-baryons are studied in the framework of the
Grand Canonical Ensemble, where the net-baryon number is conserved only on
average. Accordingly, the analysis is performed differentially as a function
of the collision centrality, the pseudorapidity interval, $\Delta\eta=$ 0.2 to
1.6, and for two different momentum ranges, 0.6–1.5 GeV/c and 0.6–2.0 GeV/c.
It should be noted that the determination of centrality and the net-proton
analysis are based on measurements in different pseudorapidity intervals to
avoid trivial effects due to autocorrelations [18].

Figure 4: (Color online) Centrality (left) and pseudorapidity interval (right)
dependence of the normalized second-order cumulants of net protons. The ALICE
data are shown by black and red markers for $\sqrt{s_{\mathrm{NN}}}=2.76$ and
$5.02$ TeV, respectively, while the colored shaded areas indicate the results
from HIJING [45] and EPOS [52] model calculations at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. The Skellam baseline is shown by the
horizontal dashed black line. In the right panel the expectation from global
baryon number conservation is shown as a pink band and the dashed colored
lines represent the predictions of the model with local baryon number
conservation [22].

Figure 4 shows the measured centrality and pseudorapidity dependence of the
normalized second-order cumulants of the net protons in Pb–Pb collisions for
the two collision energies. The 5.02 TeV data appear to be somewhat lower,
however the two data sets agree within systematic uncertainties. It should be
noted that the systematic uncertainties exhibit a large degree of correlation
from bin to bin, but between the two collision energies are essentially
uncorrelated due to the different running conditions (collision rate, gas
mixture and space charge distortions in the TPC, etc.). The normalized second-
order cumulants are independent of collision centrality and are reduced by
about 5% from the Skellam baseline for the $\Delta\eta=1.6$ interval (left
panel). As a function of the width of the $\Delta\eta$ interval, the
fluctuations are increasingly reduced. Due to the increasing relevance of
baryon number conservation with larger acceptance, this is expected. For the
narrowest interval, statistically independent Poissonian fluctuations are
observed. The results are also compared to results from HIJING [45] and EPOS
[52] model calculations at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. HIJING treats
nucleus–nucleus collisions as an independent superposition of nucleon–nucleon
interactions and does not include phenomena such as equilibrium and
collectivity. While in HIJING the hadronization is based on the Lund string
fragmentation scheme, EPOS (version 1.99, tuned to LHC data) distinguishes
between string segments in a collectively behaving central part (“core”) with
high energy density and those in a peripheral part (“corona”) with lower
energy density, more like in pp or p–A collisions.

It is noteworthy that the proton to antiproton ratio is poorly reproduced in
both models: the ratio is 1.025$\pm 0.004$ and 1.008$\pm 0.002$ for EPOS and
HIJING, respectively, while the value measured by ALICE in Pb–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV is unity within experimental uncertainties
[49]. This implies that the volume fluctuations for the second- and third-
order cumulants are not negligible for the model calculations [18].

Both model calculations show second-order fluctuations independent of
centrality, as do the experimental data. The results from the EPOS
calculations agree within the uncertainties with the data both in terms of the
centrality and the $\Delta\eta$ dependencies. The HIJING model results exhibit
a 12% suppression compared to Poissonian fluctuations for the widest
$\Delta\eta$ interval and are significantly below the data. This is also
apparent in the dependence on the width of the $\Delta\eta$ interval (right
panel of Fig. 4). The dependence on acceptance, and specifically the
discrepancy between the HIJING results and the data, is examined in view of
global vs local baryon number conservation modelled in Refs. [18, 26, 22, 51]
using a canonical statistical model. The right panel of Fig. 4 shows the
results for different widths of the correlation interval, ranging from global
baryon number conservation to short-range correlations. As expected,
measurements and model calculations converge to the Skellam baseline in the
limit of very small acceptance. As already noted in Ref. [26], the data from
ALICE indicate long-range rapidity correlations ($\Delta y_{\rm corr}>5$)
between protons and antiprotons, therefore originating from the early phase of
the collision [53]. Here $\Delta y_{\rm corr}/2$ is defined as the correlation
length between protons and antiprotons in rapidity [22], so that $\Delta
y_{\rm corr}=5$ means that protons are correlated with antiprotons within 2.5
rapidity units into either direction.

Figure 5: (Color online) Centrality (left) and pseudorapidity interval (right)
dependence of the normalized second-order cumulants of net protons for
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV and two momentum intervals for the protons.
The ALICE data are shown by red and blue markers for $0.6<p<1.5$ GeV/c and
$0.6<p<2.0$ GeV/c, respectively. The colored shaded areas indicate the results
from the HIJING [45] and EPOS [52] model calculations. In the right panel, in
addition, the dashed colored lines represent the predictions from the model
with local baryon number conservation with $\Delta y_{\rm corr}=5$ [22].

The HIJING model calculations reflect a much smaller correlation length
($\Delta y_{\rm corr}=2$) than the EPOS model ($\Delta y_{\rm corr}=5$) and
the ALICE data. This is likely due to the formation of baryons in string
breaking in the underlying Lund string model [54]. This sensitivity to the
range of proton–antiproton correlations is further studied by enlarging the
momentum acceptance for the $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV data. The
resulting normalized second-order cumulants are shown in Fig. 5 for the
momentum intervals 0.6–1.5 GeV/c and 0.6–2.0 GeV/c. Using the wider momentum
interval the number of protons and antiprotons roughly doubles. While there is
again no change with the collision centrality, the larger acceptance leads to
a larger suppression of fluctuations for the wider momentum range. For the
largest $\Delta\eta$ interval, the suppression amounts to an additional 4%.
All calculations reflect the reduction in the fluctuations. However, while the
magnitude is properly reproduced by the canonical statistical model
predictions with a long correlation length [22], it can be noted that the
suppression due to increased acceptance is somewhat weaker in the EPOS
results. The HIJING calculations properly track the absolute reduction in
fluctuations, but fall significantly below the data in absolute amount.

Figure 6 shows the centrality and pseudorapidity dependence of the ratio of
third- to second-order cumulants of net protons at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV before and after the efficiency correction.
The uncorrected results deviate from the Skellam baseline, which is set at
zero, because the net-proton number is nonzero due to antiproton absorption in
the detector material and, to a smaller degree, by a proton knock-out
contribution. Understanding and controlling the particle detection efficiency
is one of the major technical challenges in the measurement of higher-order
cumulants, since the efficiency enters into the analytical formula of the
correction with the corresponding high power [41, 42, 43]. This affects both
statistical and systematic uncertainties of the corrected data. Therefore, the
final results depend crucially on a very accurate determination of the proton
and antiproton efficiencies. Note that the efficiency correction approximately
doubles the statistical uncertainties, as also noted in Ref. [55]. The
experimentally achieved overall precision is better than 4% for the most
central collisions and much smaller for more peripheral collisions.

Figure 6: Centrality (left) and pseudorapidity interval (right) dependence of
the ratio of third- to second-order cumulants for net protons at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV before (open markers) and after (closed
markers) efficiency correction.

After efficiency correction, the data agree with the zero baseline within the
experimental uncertainties, which is consistent with expectations from the HRG
model. Note that in the HRG model all odd cumulants vanish at LHC energy,
where the number of baryons and antibaryons agree. The odd cumulants vanish
under these conditions also if baryon number conservation is included, see
Refs. [56, 51]. Also in LQCD [57] the odd cumulants vanish.

In Fig. 7, the third-order cumulant measurements are also compared with HIJING
and EPOS model calculation results. Both models include baryon number
conservation but, as mentioned above, the net-proton number is positive within
the current experimental acceptance. Therefore, the resulting third-order
cumulants for all centrality and pseudorapidity difference intervals shift
toward positive values and are affected by the volume fluctuations [18]
visible in the 10–20% centrality interval, where the centrality range doubles
(left panel). The agreement of the experimental third-order cumulants with a
value of zero is a confirmation that the average number of protons and
antiprotons is the same at LHC energies and that the systematic uncertainties
for these measurements are under good control.

Figure 7: (Color online) Centrality (left) and pseudorapidity interval (right)
dependence of the ratio of third- to second-order cumulants for net protons at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. The ALICE data are shown by red markers,
while the colored shaded bands represent the results from HIJING [45] and EPOS
[52] model calculations.

##  4 Conclusions

In summary, net-proton cumulant measurements up to third order and net-pion
and net-kaon second-order cumulant measurements are reported. The technical
challenges related to data analysis, in particular efficiency correction and
event pile-up, could be overcome as discussed in detail. Resonance
contributions prove to be challenging in the study of fluctuations of the net-
electric charge and the net-strangeness. A deviation of about 4% from the
Skellam baseline is observed for the second-order net-proton cumulants for the
widest $\Delta\eta$ interval. Investigation of this deviation in light of
baryon number conservation led to the conclusion that the 2010 data from ALICE
[26] indicate the presence of long-range rapidity correlations between protons
and antiprotons originating from the early phase of the collision. This
finding is corroborated by the present analysis including the higher
luminosity 2015 data with significantly different experimental conditions.
Results of calculations using the HIJING generator, based on the Lund string
model, reflect a much smaller correlation length of one unit of rapidity. This
observed discrepancy calls into question the mechanism implemented in the Lund
string model for the production of baryons. After accounting for the effect of
baryon number conservation, the data from ALICE are consistent with LQCD
expectations up to the third-order cumulants of the net protons. The finding
of third-order net-proton cumulants consistent with zero with a precision of
better than 4% is promising for the analysis of the higher-order cumulants
during the operation of LHC with increased Pb–Pb luminosity [58] starting in
2022 and for the future heavy-ion detector planned for the early 2030s [59].

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