# Sharp Phase Transition for the Random-Cluster Model with Summable External
Magnetic Field

R. Vila
Departamento de Estatística - Universidade de Brasília, Brazil, Email:
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove sharpness of the phase transition for the random-
cluster model in summable positive external fields, with cluster weight
$q\in\\{2,3,\ldots\\}$, on the hypercubic lattice $\mathbb{Z}^{d}$,
$d\geqslant 2$. That is, there exists some critical parameter
$0<\beta_{c}<\infty$ that depends on the cluster weight and the external
field, below which the model exhibits exponential decay and above which there
exists almost surely an infinite cluster.

###### Key words and phrases:

Random-cluster model $\cdot$ non-translation-invariant external field $\cdot$
sharp phase transition $\cdot$ exponential decay $\cdot$ FKG property.

###### 2010 Mathematics Subject Classification:

MSC 60-XX $\cdot$ MSC 60K35.

##  1\. Introduction

The random-cluster model is defined as a correlated bond percolation model for
phase transitions in lattice systems. This model was introduced by Fortuin and
Kasteleyn around 1970 [21, 22, 23] satisfying specific series and parallel
laws. The random-cluster model is a graphical representation of a range of
important models of statistical mechanics, among them the independent bond
percolation [13], the Potts model [38] and the uniform spanning trees [32].
Graphical representations are commonly used to study the existence of phase
transitions in Ising/Potts-type models relating them to the random-cluster
model through the Edwards-Sokal measure [20]. Recently, a variety of results
concerning the phase transition of the planar random-cluster model have
emerged; see [5, 15, 17, 19]. When non-translation-invariant external fields
are considered, phase transition in the random-cluster model does not always
correspond to a phase transition in the corresponding spin system. In this
case, as noticed by [6], the FKG property is harder to obtain because this
property does not even hold for general boundary conditions. Moreover, other
complications appear, for example, in the analysis of Dobrushin-like states
[25] and the effect of weak boundary conditions in the Potts model [10].
Graphical representations [13] for the Potts model in external fields have
been the object of revived interest after rigorous results were proved; see
[6, 11, 12].

This paper is motivated by some recent works on ferromagnetics spin systems in
non-uniform external fields; see [4, 7, 8, 9, 11, 30, 36, 37]. The objective
of this paper is to prove that the phase transition for the random-cluster
model over the hypercubic lattice, allowing a summable positive external
field, is sharp. Roughly speaking, by using the strong FKG property and
comparison inequalities, we showed that there is a certain non-trivial
critical point, $\beta_{c}$ above which there is almost surely a path of
connected points of infinite length through the network. Below it, the model
exhibits exponential decay of the infinite-volume connectivity function for
two points. Sufficient conditions for the (uniform) exponential decay of
finite-volume connectivities in the planar random-cluster model, with external
field, was established in [3]. In the absence of an external magnetic field,
sharp phase transitions for the random-cluster model, Potts model and related
percolation models, on several different types of graphs, have been studied by
[1, 2, 14, 16, 18, 33].

This paper is organized as follows. In Section 2, we present the random-
cluster model with free and max-wired boundary conditions, and some
definitions related to this model. We close this section by stating the main
result of paper. In Section 3, we present some preliminary monotonicities of
random-cluster measures. Finally, in Section 4, we present the proof of the
main result in details.

##  2\. The model and the main result

In this section, we introduce the random-cluster model and state the main
result of the paper. For general results and historical background of the
random-cluster model, we refer the reader to [6, 11, 13, 26] and the
references therein.

The model is defined as follows. Consider a finite subgraph $G=(V,E)$ of the
infinite countable connected graph
$\mathbb{G}=(\mathbb{Z}^{d},\mathbb{E}^{d})$, where $\mathbb{E}^{d}$ is the
set of the nearest-neighbor edges $xy$ in the $d$-dimensional hypercubic
lattice $\mathbb{Z}^{d}$, with $d\geqslant 2$. The boundaries of $V$ and $E$
are given by $\partial V=\\{x\notin V:xy\in\mathbb{E}^{d}\ \text{for some}\
y\in V\\}$ and $\partial E=\\{xy\in\mathbb{E}^{d}:xy\cap V\neq\emptyset\
\text{and}\ xy\cap\partial V\neq\emptyset\\}$, respectively. The configuration
space of the random-cluster model is the product space
$\\{0,1\\}^{E\cup\partial E}$. A generic element of this space, denoted by
$\omega=(\omega_{xy})_{{xy}\in E\cup\partial E}$, is often called edge
configuration. We say that an edge $xy$ is open in a configuration $\omega$ if
$\omega_{xy}=1$, and closed otherwise. Given
$\omega\in\\{0,1\\}^{E\cup\partial E}$, we set $x\longleftrightarrow y$ in
$\omega$ if $x$ and $y$ are in the same connected component of $\omega$. The
random sets $\eta(\omega)=\\{xy\in E\cup\partial E:\omega_{xy}=1\\}$ and
$C_{x}(\omega)=\\{y\in V\cup\partial V:x\longleftrightarrow y\ \text{in}\
\omega\\}\cup\\{x\\}$ denote the set of open edges and the open cluster of a
vertex $x\in V$ in $\omega$, respectively. We write $|C_{x}|$ to denote the
number of vertices in $C_{x}$.

We fix two families
$\bm{J}=(J_{xy})_{xy\in\mathbb{E}^{d}}\in[0,\infty)^{\mathbb{E}^{d}}$ and
$\bm{\widehat{h}}=(h_{x,m})_{x\in{\mathbb{Z}^{d}},m\in\\{1,\ldots,q\\}}\in(\mathbb{R}^{\mathbb{Z}^{d}})^{q}$
called coupling constant and external field, respectively, where
$q\in\\{2,3,\ldots\\}$. Throughout this paper, we define
$h_{x,\mathrm{max}}=\max_{m\in\\{1,\ldots,q\\}}h_{x,m},$ for each vertex $x\in
V$, denote

(1)
$\displaystyle{Q}_{x,\mathrm{max}}(\bm{\widehat{h}})=\big{\\{}m\in\\{1,\ldots,q\\}:h_{x,m}=h_{x,\mathrm{max}}\big{\\}},$

and assume that

(2)
$\displaystyle\left|{\cap}_{x\in{V}}{Q}_{x,\mathrm{max}}(\bm{\widehat{h}})\right|\geqslant
1.$

Fix $\beta>0$ and ${\rm m}\in\\{1,\ldots,q\\}$. For $\\#\in\\{{\rm f},{\rm
m}\\}$, following the references [6, 11], let
$\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}$ be the measure satisfying, for any
$\omega\in\\{0,1\\}^{E\cup\partial E}$,

(3) $\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}(\omega)={1\over
Z^{\\#}_{G;\beta,q,\bm{\widehat{h}}}}\,\prod_{xy\in E}\big{[}\exp({q\beta
J_{xy}})-1\big{]}^{\omega_{xy}}\,\prod_{C(\omega)}\Theta^{\\#}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]},$

where $Z^{\\#}_{G;\beta,q,\bm{\widehat{h}}}$ is a normalizing constant, the
second product is over all open clusters of sites, and the weights of the
clusters are given by

$\Theta^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]}=\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
C(\omega)}h_{x,m}\Bigg{)},$

for all open clusters $C(\omega)$ of the graph $\big{(}V,\eta(\omega)\big{)}$
with $\omega\in\\{0,1\\}^{E}$ (since $\omega_{xy}=0$ for all $xy\in\partial
E$); and

$\displaystyle\Theta^{\rm
m}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]}=\begin{cases}\Theta^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]},&\text{if}\
C(\omega)\cap\partial V=\emptyset,\\\\[11.38092pt]
\displaystyle\exp\Bigg{(}\beta\sum_{x\in C(\omega)}h_{x,{\rm
m}}\Bigg{)},&\text{otherwise},\end{cases}$

for all open clusters $C(\omega)$ of the graph $\big{(}V\cup\partial
V,\eta(\omega)\big{)}$ with $\omega\in\\{0,1\\}^{E\cup\partial E}$, where
$\omega_{xy}=1$ for all $xy\in\partial E$. If ${\rm m},\widetilde{{\rm
m}}\in{\cap}_{x\in{V}}{Q}_{x,\mathrm{max}}(\bm{\widehat{h}}),$ then

$\displaystyle\Theta^{\rm
m}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]}=\Theta^{\widetilde{{\rm
m}}}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]}\quad\text{for all}\
\omega\in\\{0,1\\}^{E\cup\partial E},$

and therefore $\phi^{\rm m}_{G;\beta,q,\bm{\widehat{h}}}=\phi^{\widetilde{{\rm
m}}}_{G;\beta,q,\bm{\widehat{h}}}.$ This measure is denoted by $\phi^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}$. The measures $\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}$ and $\phi^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}$ are called the random-cluster measures on $G$
with free and max-wired boundary conditions, respectively. By taking the weak
limit of measures defined in finite volume, the measures
$\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}$, with $\\#\in\\{{\rm f},{\rm w}\\}$,
can be extended to $\mathbb{G}$; see Theorem 3.3. The expected value of a
real-valued function $f:\\{0,1\\}^{E\cup\partial E}\to\mathbb{R}$ in the
random-cluster model is denoted by
$\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}(f)$. When $f=\mathds{1}_{A}$, for
some cylindrical event $A$, we write
$\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}(A)$ instead
$\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}(\mathds{1}_{A}).$

The functions of principal interest are the following:

(4)
$\displaystyle\theta(\beta,q,\bm{\widehat{h}})\coloneqq\sup_{x\in\mathbb{Z}^{d}}\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|=\infty\big{)};\quad\chi(\beta,q,\bm{\widehat{h}})\coloneqq\sup_{x\in\mathbb{Z}^{d}}\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|\big{)}.$

These functions are known as percolation-probability and susceptibility
function, respectively.

We write $\bm{\widehat{h}}\in\ell^{1}(\mathbb{Z}^{d})$ when $\bm{\widehat{h}}$
is summable, that is,
$\lVert\bm{\widehat{h}}\rVert_{1}=\sum_{m=1}^{q}\sum_{x\in\mathbb{Z}^{d}}|h_{x,m}|<\infty$.
The following theorem is the main result of paper. This result discusses the
existence of phase transition and the exponential decay of the infinite-volume
connectivity function for two points.

######  Theorem 1.

Fix $q\in\\{2,3,\ldots\\}$ and $d\geqslant 2$. Consider the random-cluster
model on $\mathbb{G}$, with uniform coupling constant $\bm{J}$, i.e.
$J_{xy}=J\geqslant 0$ is constant for all $xy\in\mathbb{E}^{d}$, and external
field
$\bm{\widehat{h}}=(h_{x,m})_{x\in\mathbb{Z}^{d},m\in\\{1,\ldots,q\\}}\in(\mathbb{R}^{\mathbb{Z}^{d}})^{q}$
satisfying the condition (2). Then, there exists a critical point
$0<\beta_{c}=\beta_{c}(q,\bm{\widehat{h}})<\infty$ so that

  * 1)

$\theta(\beta,q,\bm{\widehat{h}})=0$ if $\beta<\beta_{c}$, and
$\theta(\beta,q,\bm{\widehat{h}})>0$ if $\beta>\beta_{c}$.

  * 2)

$\chi(\beta,q,\bm{\widehat{h}})<\infty$ if $\beta<\beta_{0},$ for some
$0<\beta_{0}<\beta_{c}$; and $\chi(\beta,q,\bm{\widehat{h}})=\infty$ if
$\beta>\beta_{c}$.

  * 3)

In the case that $\bm{\widehat{h}}\in\ell^{1}(\mathbb{Z}^{d})$ has positive
terms, for all $\beta<\beta_{c}$ there exist constants
$C(\beta,\bm{\widehat{h}}),\gamma(\beta,q)>0$, such that

$\displaystyle\displaystyle\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow y)\leqslant
C(\beta,\bm{\widehat{h}})\,{\exp}\big{[}{-\|x-y\|\gamma(\beta,q)}\big{]}\quad\text{for
all}\ x,y\in\mathbb{Z}^{d},$

where $\|\cdot\|$ denotes the Euclidean norm. This implies finite
susceptibility.

This theorem extends to a large class of amenable infinite transitive graphs.

As a consequence of Theorem 1 we have the sharpness of the phase transition
for the random-cluster model in summable positive external fields, on the
hypercubic lattice. Remains to investigate the following. For which classes of
non-summable external fields, the phase transition of the random-cluster on
the graph $\mathbb{G}$ is sharp? Sharp phase transitions for spin systems and
related percolation models, in the absence of an external magnetic field, on
different types of graphs have emerged in references [1, 2, 14, 16, 18, 33].

The rest of this paper is devoted to the proof of Theorem 1. To this end, some
comparison inequalities are essential.

##  3\. Monotonic measures

Let $G=(V,E)$ be a finite subgraph of $\mathbb{G}$ and $q\in\\{2,3,\ldots\\}$.
We consider the usual partial order on $\\{0,1\\}^{E\cup\partial E}$ where
$\omega\preceq\tilde{\omega}$ iff $\omega_{xy}\leqslant\tilde{\omega}_{xy}$
for all $xy\in E\cup\partial E$.

######  Definition 3.1.

Let $(\Omega,\preceq)$ be a partially ordered space. A measure $\mu$ over
$\Omega$ said to have the $\mathrm{FKG}$ property if

$\mu(fg)\geqslant\mu(f)\mu(g),$

for any increasing (with respect to $\preceq$) measurable functions
$f,g:\Omega\to\mathbb{R}$. Furthermore, if $\Omega$ is a cartesian product
$\Omega=\prod_{xy\in B}\Omega_{xy}$, with $|\Omega_{xy}|<\infty$, then $\mu$
is said to have the strong FKG property, if $\mu(\cdot|A)$ has the
$\mathrm{FKG}$ property for each cylinder event
$A=\\{\omega\in\Omega:\omega_{xy}=\alpha_{xy}\ \text{for all}\
xy\in\\{0,1\\}^{E\cup\partial E}\in\widetilde{B}\\}$, where
$\widetilde{B}\subset{B}$ is finite and $\alpha_{xy}\in\Omega_{xy}$ for all
$xy\in\widetilde{B}$.

######  Theorem 3.2 ([11]).

The finite-volume measures $\phi^{\rm f}_{G;\beta,q,\bm{\widehat{h}}}$ and
$\phi^{\rm w}_{G;\beta,q,\bm{\widehat{h}}}$ have the strong $\mathrm{FKG}$
property.

By using the strong FKG property for the random-cluster model, one can prove
the following two theorems.

######  Theorem 3.3 ([11]).

For each increasing quasilocal function f (see [24]),

  * (a)

The following limits exist

$\displaystyle\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(f)=\lim_{G\uparrow\mathbb{G}}\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}(f);\quad\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(f)=\lim_{G\uparrow\mathbb{G}}\phi^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}(f).$

  * (b)

The finite-volume measures $\phi^{\rm f}_{{G};\beta,q,\bm{\widehat{h}}}$ and
$\phi^{\rm w}_{{G};\beta,q,\bm{\widehat{h}}}$ are the extremal ones, in the
sense that, if $\phi_{{G};\beta,q,\bm{\widehat{h}}}$ is a finite-volume
random-cluster measure, then

$\displaystyle\phi^{\rm
f}_{{G};\beta,q,\bm{\widehat{h}}}(f)\leqslant\phi_{{G};\beta,q,\bm{\widehat{h}}}(f)\leqslant\phi^{\rm
w}_{{G};\beta,q,\bm{\widehat{h}}}(f).$

Consequently, $\phi^{\rm f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}$ and
$\phi^{\rm w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}$ are also extremal
measures.

Hereafter $N_{\infty}$ denotes the random variable that counts the number of
infinite open clusters in the sample space $\\{0,1\\}^{\mathbb{E}^{d}}$. The
following theorem states that the probability that a unique infinite cluster
exists is either zero or one.

######  Theorem 3.4 ([11]).

The measures $\phi^{\\#}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}$, with
$\\#\in\\{{\rm f},{\rm w}\\}$, have the almost-sure uniqueness of the infinite
open cluster property. That is,

$\displaystyle\phi^{\\#}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(N_{\infty}\leqslant
1)=1\quad\text{for}\ \\#\in\\{{\rm f},{\rm w}\\}.$

The next monotonicity result is valid for a large class of probability
measures with the restriction that these measures are positively associated.

######  Proposition 3.5.

Let $(\Omega,\mathscr{F},\mu_{i})$, $i=1,2,$ two probability spaces with
probability measures
$\mu_{i}(\omega)=\mathcal{K}(\omega)\mathcal{W}_{i}(\omega)/Z_{i}$,
$\omega\in\Omega$, $i=1,2,$ satisfying the FKG property, where $Z_{i}$ is the
respective partition function and $\mathcal{K}$ is a non-negative function. If
$\mathcal{W}_{1}/\mathcal{W}_{2}$ is an increasing non-negative function, then
$\mu_{1}(f)\geqslant\mu_{2}(f)$ for each cylindrical increasing function $f$.

###### Proof.

Let $g(\omega)={\mathcal{W}_{1}(\omega)/\mathcal{W}_{2}(\omega)}$,
$\omega\in\Omega$. Note that $g$ is an increasing function. A simple
computation shows that, for all cylindrical valued-real function $f$,

$\displaystyle\mu_{1}(f)={Z_{2}\over Z_{1}}\,\mu_{2}(fg).$

Taking $f\equiv 1$ we obtain that $\mu_{2}(g)={Z_{1}/Z_{2}}.$ Then, by using
the above identity, from FKG inequality we have, for each cylindrical
increasing function $f$,

$\mu_{1}(f)={\mu_{2}(fg)\over\mu_{2}(g)}\geqslant\mu_{2}(f).$

Thus the proof is complete. ∎

The next lemma shows that the part of the measures
$\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}$ in (3), with $\\#\in\\{{\rm f},{\rm
w}\\}$, due to the external magnetic field are decreasing functions.

######  Lemma 3.6.

Under the condition (2), the following hold:

  * (a)

The function ${g}^{\rm f}$ defined as, for any $\omega\in\\{0,1\\}^{E}$,

${g}^{\rm f}(\omega)\coloneqq\prod_{C(\omega)}\Theta^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]}=\prod_{C(\omega)}\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
C(\omega)}h_{x,m}\Bigg{)},$

is monotone decreasing in the FKG sense, where the above product runs over all
the open clusters $C(\omega)$ of the graph $\big{(}V,\eta(\omega)\big{)}$.

  * (b)

The function ${g}^{\rm w}$ defined as, for any
$\omega\in\\{0,1\\}^{E\cup\partial E}$,

$\displaystyle{g}^{\rm w}(\omega)$
$\displaystyle\coloneqq\prod_{C(\omega)}\Theta^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]}$
$\displaystyle=\prod_{C(\omega):C(\omega)\cap\partial
V=\emptyset}\,\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
C(\omega)}h_{x,m}\Bigg{)}\prod_{C(\omega):C(\omega)\cap\partial
V\neq\emptyset}\exp\Bigg{(}\beta\sum_{x\in C(\omega)}h_{x,{\rm max}}\Bigg{)},$

is monotone decreasing in the FKG sense.

###### Proof.

Let $xy$ be a nearest-neighbor edge such that $\omega_{xy}=0$ and let
$\omega^{xy}$ be the configuration obtained by flipping $\omega_{xy}$ to $1$.
It suffices to study the single-bond flips. That is, it is sufficient to prove
the following

(5)
$\displaystyle{g}^{\\#}(\omega^{xy})\leqslant{g}^{\\#}(\omega)\quad\text{with}\
\\#\in\\{{\rm f},{\rm w}\\}.$

If the vertices $x$ and $y$ are connected in the configuration $\omega$,
${g}^{\\#}(\omega^{xy})={g}^{\\#}(\omega)$. If the vertices $x$ e $y$ are not
connected in $\omega$, then there are two open clusters $A=C_{x}(\omega)$ and
$B=C_{y}(\omega)$ containing the vertices $x$ and $y$, respectively. If $xy$
is an open edge at $\omega$, then the clusters $A$ and $B$ are connected,
creating a new open cluster denoted by $C=A\cup B$. Then, $|C|=|A|+|B|,$ and
so to prove (5), it is enough to verify that

(6)
$\displaystyle\Theta^{\\#}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C\big{]}\leqslant\Theta^{\\#}_{G;\beta,q,\bm{\widehat{h}}}\big{[}A\big{]}\Theta^{\\#}_{G;\beta,q,\bm{\widehat{h}}}\big{[}B\big{]}\quad\text{with}\
\\#\in\\{{\rm f},{\rm w}\\}.$

(a) When $\\#={\rm f}$, the inequality in (6) can be rewritten as

$\displaystyle\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
C}h_{x,m}\Bigg{)}\leqslant\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
A}h_{x,m}\Bigg{)}\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in B}h_{x,m}\Bigg{)}.$

But the above inequality is true since

$\sum_{k=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
A}h_{x,k}\Bigg{)}\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
B}h_{x,m}\Bigg{)}\\\\[5.69046pt]
\geqslant\sum_{k=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
A}h_{x,k}\Bigg{)}\exp\Bigg{(}\beta\sum_{x\in
B}h_{x,\mathrm{max}}\Bigg{)}\left|{\cap}_{x\in{V}}{Q}_{x,\mathrm{max}}(\bm{\widehat{h}})\right|\\\
\stackrel{{\scriptstyle}}{{\geqslant}}\sum_{j=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
C}h_{x,j}\Bigg{)},$

where in the last line we have used the condition (2).

(b) When $\\#={\rm w}$, we analyze the following possible cases. (i)
$A\cap\partial V=\emptyset$ and $B\cap\partial V=\emptyset$; (ii)
$A\cap\partial V\neq\emptyset$ and $B\cap\partial V=\emptyset$; (iii)
$A\cap\partial V=\emptyset$ and $B\cap\partial V\neq\emptyset$; and (iv)
$A\cap\partial V\neq\emptyset$ and $B\cap\partial V\neq\emptyset$.

In the case (i), the inequality in (6) follows directly from Item (a). In the
case (ii), (6) is equivalent to the following inequality

$\displaystyle\exp\Bigg{(}\beta\sum_{x\in C}h_{x,{\rm
max}}\Bigg{)}\leqslant\exp\Bigg{(}\beta\sum_{x\in A}h_{x,{\rm
max}}\Bigg{)}\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in B}h_{x,m}\Bigg{)}.$

But this is true since

$\exp\Bigg{(}\beta\sum_{x\in A}h_{x,{\rm
max}}\Bigg{)}\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
B}h_{x,m}\Bigg{)}\\\\[2.84544pt] \geqslant\exp\Bigg{(}\beta\sum_{x\in
A}h_{x,{\rm max}}\Bigg{)}\exp\Bigg{(}\beta\sum_{x\in
B}h_{x,\mathrm{max}}\Bigg{)}\left|{\cap}_{x\in{V}}{Q}_{x,\mathrm{max}}(\bm{\widehat{h}})\right|\\\
\geqslant\exp\Bigg{(}\beta\sum_{x\in C}h_{x,{\rm max}}\Bigg{)},$

where, again, in the last line we have used the condition (2). In the case
(iii), the inequality in (6) is proven analogously to Item (ii). Finally, in
the case (iv), (6) is equivalent to the following identity

$\displaystyle\exp\Bigg{(}\beta\sum_{x\in C}h_{x,{\rm
max}}\Bigg{)}=\exp\Bigg{(}\beta\sum_{x\in A}h_{x,{\rm
max}}\Bigg{)}\exp\Bigg{(}\beta\sum_{x\in B}h_{x,{\rm max}}\Bigg{)}.$

Hence the proof follows. ∎

It is known that stochastic dominance allows to compare different probability
measures on the same sample space. One of the common uses of this type of
comparisons is to prove that certain characteristics are inherited from one
random model to another, which is generally not trivial. The following lemma
states that, at finite volume, the Bernoulli measure dominates the random-
cluster measure with external field. Hence, for example, we can use this lemma
to prove that, at the thermodynamic limit, the almost-sure uniqueness (under
amenabilitity condition in the infinite graph) of the infinite open cluster
would be inherited for the infinite-volume random-cluster measure; see Theorem
3.4 and Theorem 10 in [11] for a rigorous proof of this fact.

######  Lemma 3.7.

For each cylindrical increasing function $f$ we have

$\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}(f)\leqslant\phi^{\\#}_{G;\beta,1,\bm{\widehat{h}}}(f)=\phi^{\\#}_{G;\beta,1,\bm{\widehat{0}}}(f)\eqqcolon\mathbb{P}_{G;\beta}(f)\quad\text{with}\
\\#\in\\{{\rm f},{\rm w}\\},$

where $\mathbb{P}_{G;\beta}$ becomes the product probability measure on
$\\{0,1\\}^{E}$ defined by the Bernoulli factor $\prod_{xy\in
E}p_{xy}^{\omega_{xy}}(1-p_{xy})^{1-\omega_{xy}}$, with
$p_{xy}=1-\exp({-q\beta J_{xy}})$.

###### Proof.

Let $\mu_{1}$ and $\mu_{2}$ be two probability measures defined by
$\mathbb{P}_{G;\beta}$ and $\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}$,
respectively. The respective weights of these measures are given by
$\mathcal{W}_{1}(\omega)=\exp\big{(}{-q\beta\sum_{xy\in E}J_{xy}}\big{)}$,
$\mathcal{W}_{2}(\omega)={g}^{\\#}(\omega)$ and
$\mathcal{K}(\omega)=\prod_{xy\in E}\big{[}\exp({q\beta
J_{xy}})-1\big{]}^{\omega_{xy}}$, $\omega\in\\{0,1\\}^{E}$, where ${g}^{\rm
f}$ and ${g}^{\rm w}$ are given in Lemma 3.6. Since $\mathcal{W}_{2}$ is
decreasing (see Lemma 3.6), $\mathcal{W}_{1}/\mathcal{W}_{2}$ is an increasing
function. Hence, by Proposition 3.5, we have that
$\mu_{1}(f)\geqslant\mu_{2}(f)$ for each cylindrical increasing function $f$.

On the other hand, the identity
$\phi^{\\#}_{G;\beta,1,\bm{\widehat{h}}}(f)=\mathbb{P}_{G;\beta}(f)$ follows
immediately by taking $q=1$ in (3). ∎

In what follows we consider as usual the partial order on $[0,\infty)^{E}$
where $\bm{J}\curlyeqprec\bm{J}^{\prime}$ iff $J_{xy}\leqslant
J_{xy}^{\prime}$ for all $xy\in E.$

######  Proposition 3.8.

Suppose that $\bm{J}\curlyeqprec\bm{J}^{\prime}$ are two coupling constants.
The function $\widetilde{g}$ defined as, for any $\omega\in\\{0,1\\}^{E}$,

$\widetilde{g}(\omega)\coloneqq{\prod_{xy\in E}\big{[}\exp({q\beta
J_{xy}})-1\big{]}^{\omega_{xy}}\over\prod_{xy\in E}\big{[}\exp({q\beta
J_{xy}^{\prime}})-1\big{]}^{\omega_{xy}}},$

is monotone decreasing in the FKG sense.

###### Proof.

Let $x_{0}y_{0}$ be a nearest-neighbor edge such that $\omega_{x_{0}y_{0}}=0$
and let $\omega^{x_{0}y_{0}}$ be the configuration obtained by flipping
$\omega_{x_{0}y_{0}}$ to $1$. To prove the monotonicity of $\widetilde{g}$ it
is sufficient to prove that
$\widetilde{g}(\omega^{x_{0}y_{0}})\leqslant\widetilde{g}(\omega)$. But this
is true because

$\displaystyle{\widetilde{g}(\omega^{x_{0}y_{0}})\over\widetilde{g}(\omega)}$
$\displaystyle={\prod_{xy\in E}\big{[}{\exp}({q\beta
J_{xy}})-1\big{]}^{\omega^{x_{0}y_{0}}_{xy}}\over\prod_{xy\in
E}\big{[}{\exp}({q\beta
J_{xy}^{\prime}})-1\big{]}^{\omega^{x_{0}y_{0}}_{xy}}}\,{\prod_{xy\in
E}\big{[}{\exp}({q\beta
J_{xy}^{\prime}})-1\big{]}^{\omega_{xy}}\over\prod_{xy\in
E}\big{[}{\exp}({q\beta J_{xy}})-1\big{]}^{\omega_{xy}}}$
$\displaystyle={{\exp}({q\beta J_{x_{0}y_{0}}})-1\over{\exp}({q\beta
J_{x_{0}y_{0}}^{\prime}})-1}\leqslant 1,$

whenever $\bm{J}\curlyeqprec\bm{J}^{\prime}$. Then the proof follows. ∎

Our next result is the monotonicity, in the FKG sense, with respect to the
coupling constants. The first item of this result is proved in reference [11]
when the coupling constant is uniform.

######  Proposition 3.9.

Denote by $\phi^{{\\#},\bm{J}}_{G;\beta,q,\bm{\widehat{h}}}$ the random-
cluster measure on $G$ with $\\#\in\\{{\rm f},{\rm w}\\}$ boundary condition
and with coupling constant $\bm{J}=(J_{xy})_{xy\in{E}}$ $\in[0,\infty)^{E}$.
Suppose that $\bm{J}\curlyeqprec\bm{J}^{\prime}$ are two coupling constants.
Then, for any cylindrical increasing function $f$,

  * (a)

$\displaystyle\phi^{{\\#},\bm{J}}_{G;\beta,q,\bm{\widehat{h}}}(f)\leqslant\phi^{{\\#},\bm{J}^{\prime}}_{G;\beta,q,\bm{\widehat{h}}}(f).$

  * (b)

In the particular case that $\bm{J}$ is an uniform coupling constant,

    * (i)

the mapping $\beta\longmapsto\phi^{{\\#}}_{G;\beta,q,\bm{\widehat{h}}}(f)$ is
increasing;

    * (ii)

the mappings $\beta\longmapsto\theta(\beta,q,\bm{\widehat{h}})$ and
$\beta\longmapsto\chi(\beta,q,\bm{\widehat{h}})$ are increasing.

###### Proof.

In order to prove Item (a), we only present the argument for $\\#={\rm w}$,
since for the free boundary condition case the proof works similarly.

Let $\mu_{1}$ and $\mu_{2}$ be two probability measures defined by $\phi^{{\rm
w},\bm{J}^{\prime}}_{G;\beta,q,\bm{\widehat{h}}}$ and $\phi^{{\rm
w},\bm{J}}_{G;\beta,q,\bm{\widehat{h}}}$, respectively. The respective weights
of these measures are given by, for any $\omega\in\\{0,1\\}^{E\cup\partial
E}$,

$\displaystyle\mathcal{W}_{1}(\omega)=\prod_{xy\in E}\big{[}\exp({q\beta
J_{xy}^{\prime}})-1\big{]}^{\omega_{xy}}\,{g}^{\rm w}(\omega);$
$\displaystyle\mathcal{W}_{2}(\omega)=\prod_{xy\in E}\big{[}\exp({q\beta
J_{xy}})-1\big{]}^{\omega_{xy}}\,{g}^{\rm w}(\omega);$

and $\mathcal{K}(\omega)=1$, where the function ${g}^{\rm w}$ is given in
Lemma 3.6. Since $\bm{J}\curlyeqprec\bm{J}^{\prime}$, by Proposition 3.8, the
function $\mathcal{W}_{2}/\mathcal{W}_{1}$, given by

$\displaystyle{\mathcal{W}_{2}(\omega)\over\mathcal{W}_{1}(\omega)}={\prod_{xy\in
E}\big{[}\exp({q\beta J_{xy}})-1\big{]}^{\omega_{xy}}\over\prod_{xy\in
E}\big{[}\exp({q\beta
J_{xy}^{\prime}})-1\big{]}^{\omega_{xy}}}=\widetilde{g}(\omega),$

is monotone decreasing in the FKG sense. Therefore,
$\mathcal{W}_{1}/\mathcal{W}_{2}$ is an increasing function. Hence, by
Proposition 3.5, we have that $\mu_{1}(f)\geqslant\mu_{2}(f)$ for each
cylindrical increasing function $f$. This proves the inequality of the first
item.

The statements in Item (b) follow by direct application of Item (a). So we
have finished the proof. ∎

Since we are also interested in monotonicity properties with respect to the
magnetic field, it is needed to introduce a partial order between two fields
as in reference [11]. Given two arbitrary magnetic fields
$\bm{\widehat{h}}=(h_{x,m})_{x\in{V},m\in\\{1,\ldots,q\\}}$ and
$\bm{\widehat{h}}^{\prime}=(h_{x,m}^{\prime})_{x\in{V},m\in\\{1,\ldots,q\\}}$
in $(\mathbb{R}^{V})^{q}$, we say that

(7) $\displaystyle\bm{\widehat{h}}\prec\bm{\widehat{h}}^{\prime}\
\Longleftrightarrow\ \forall\ x\in V:\ h_{x,k}-h_{x,l}\leqslant
h^{\prime}_{x,k}-h^{\prime}_{x,l},\ \ \ k,l=1,\ldots,q,$

whenever $h_{x,k}-h_{x,l}>0.$

######  Theorem 3.10 ([11]).

Let $\bm{\widehat{h}}\prec\bm{\widehat{h}}^{\prime}$ be two arbitrary magnetic
fields in $(\mathbb{R}^{V})^{q}$. Then, for any quasilocal increasing function
$f$,

$\displaystyle\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}(f)\leqslant\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}^{\prime}}(f)\quad\text{with}\
\\#\in\\{{\rm f},{\rm w}\\}.$

The following result is a direct application of the above theorem.

######  Corollary 3.11.

If $\bm{\widehat{h}}$ is an arbitrary external field in
$(\mathbb{R}^{V})^{q}$, then, for any cylindrical increasing function $f$,

$\displaystyle\phi^{\\#}_{G;\beta,q,\bm{\widehat{0}}}(f)\leqslant\phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}(f)\quad\text{with}\
\\#\in\\{{\rm f},{\rm w}\\}.$

###### Proof.

Given $\varepsilon>0$ small enough, we consider the non-uniform external
magnetic field $\varepsilon\bm{\widehat{h}}\coloneqq(\varepsilon
h_{x,m})_{x\in{V},m\in\\{1,\ldots,q\\}}$. A straightforward computation shows
that $\varepsilon\bm{\widehat{h}}\prec\bm{\widehat{h}}$, where the partial
order ‘$\prec$’ is given in (7). Then, by Theorem 3.10, it follows that

$\displaystyle\phi_{G;\beta,q,\varepsilon\bm{\widehat{h}}}^{\\#}(f)\leqslant\phi_{G;\beta,q,\bm{\widehat{h}}}^{\\#}(f)\quad\text{with}\
\\#\in\\{{\rm f},{\rm w}\\},$

for each cylindrical increasing function $f$. Taking $\epsilon\to 0^{+}$ in
this inequality we get that
$\phi_{G;\beta,q,\bm{\widehat{0}}}^{\\#}(f)\leqslant\phi_{G;\beta,q,\bm{\widehat{h}}}^{\\#}(f).$
We thus complete the proof. ∎

The next result compares the random-cluster measure in summable positive
external fields with the random-cluster measure of their counterparts under
the zero external field.

######  Lemma 3.12.

If $\bm{\widehat{h}}\in\ell^{1}(\mathbb{Z}^{d})$ is a summable external field
with positive terms, there exists $C(\beta,\bm{\widehat{h}})>0$ such that, for
any non-negative cylindrical function $f$,

$\displaystyle\ \phi^{\\#}_{G;\beta,q,\bm{\widehat{h}}}(f)\leqslant
C(\beta,\bm{\widehat{h}})\,\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{0}}}(f)\quad\text{with}\ \\#\in\\{{\rm f},{\rm
w}\\}.$

###### Proof.

We only present the argument for $\\#={\rm w}$, since for the free boundary
condition case the proof works similarly. Indeed, for all open cluster
$C(\omega)$ such that $C(\omega)\cap\partial V=\emptyset$, since $h_{x,{\rm
max}}>h_{x,m}$ for all $x$ and $m$, we have

$\displaystyle\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
C(\omega)}h_{x,m}\Bigg{)}\leqslant q\bm{\cdot}\exp\Bigg{(}\beta\\!\sum_{x\in
C(\omega)}h_{x,{\rm max}}\Bigg{)}.$

This implies that, for any open cluster $C(\omega)$ of the graph
$\big{(}V\cup\partial V,\eta(\omega)\big{)}$ with
$\omega\in\\{0,1\\}^{E\cup\partial E}$,

$\displaystyle\prod_{C(\omega)}\Theta^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]}$
$\displaystyle=\prod_{C(\omega):C(\omega)\cap\partial
V=\emptyset}\,\sum_{m=1}^{q}\exp\Bigg{(}\beta\sum_{x\in
C(\omega)}h_{x,m}\Bigg{)}\prod_{C(\omega):C(\omega)\cap\partial
V\neq\emptyset}\exp\Bigg{(}\beta\sum_{x\in C(\omega)}h_{x,{\rm max}}\Bigg{)}$
(8) $\displaystyle\leqslant
q^{\big{|}\big{\\{}C(\omega):\,C(\omega)\cap\partial
V=\emptyset\big{\\}}\big{|}}\,\exp\Bigg{(}\beta\sum_{x\in V\cup\partial
V}h_{x,{\rm max}}\Bigg{)}.$

Since $\bm{\widehat{h}}$ has positive terms, $Z^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}\geqslant Z^{\rm
f}_{G;\beta,q,\bm{\widehat{0}}}\,\exp(\beta\sum_{x\in\partial V}h_{x,{\rm
max}})$. Hence, by combining this inequality with the one in (8), it follows
that, for any non-negative cylindrical function $f$,

${1\over Z^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}}\,\sum_{\omega}f(\omega)\prod_{xy\in
E}\big{[}\exp({q\beta
J_{xy}})-1\big{]}^{\omega_{xy}}\,\prod_{C(\omega)}\Theta^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}\big{[}C(\omega)\big{]}\\\\[4.26773pt]
\leqslant{\exp\big{(}\beta\sum_{x\in V}h_{x,{\rm max}}\big{)}\over Z^{\rm
f}_{G;\beta,q,\bm{\widehat{0}}}}\,\sum_{\omega}f(\omega)\prod_{xy\in
E}\big{[}\exp({q\beta
J_{xy}})-1\big{]}^{\omega_{xy}}\,q^{\big{|}\big{\\{}C(\omega):\,C(\omega)\cap\partial
V=\emptyset\big{\\}}\big{|}}\\\\[4.26773pt]
\leqslant{{\exp}\big{(}{\beta\lVert\bm{\widehat{h}}\rVert_{1}}\big{)}\over
Z^{\rm f}_{G;\beta,q,\bm{\widehat{0}}}}\,\sum_{\omega}f(\omega)\prod_{xy\in
E}\big{[}\exp({q\beta
J_{xy}})-1\big{]}^{\omega_{xy}}\,q^{\big{|}\big{\\{}C(\omega):\,C(\omega)\cap\partial
V=\emptyset\big{\\}}\big{|}},$

because

$\displaystyle\exp\Bigg{(}\beta\sum_{x\in V}h_{x,{\rm
max}}\Bigg{)}\leqslant\exp\Bigg{(}\beta\sum_{x\in\mathbb{Z}^{d}}\max_{m\in\\{1,\ldots,q\\}}|h_{x,m}|\Bigg{)}\leqslant{\exp}\big{(}{\beta\lVert\bm{\widehat{h}}\rVert_{1}}\big{)}.$

Therefore, $\phi^{\rm
w}_{G;\beta,q,\bm{\widehat{h}}}(f)\leqslant{\exp}\big{(}{\beta\lVert\bm{\widehat{h}}\rVert_{1}}\big{)}\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{0}}}(f)$. Hence, the validity of lemma, with
$\\#={\rm w}$, follows by taking
$C(\beta,\bm{\widehat{h}})={\exp}\big{(}{\beta\lVert\bm{\widehat{h}}\rVert_{1}}\big{)}$.
∎

##  4\. Proof of Theorem 1

In this section we apply the results obtained in the last section to prove the
main result of paper and to establish some remarks concerning the behavior of
the random-cluster model on the graph
$\mathbb{G}=(\mathbb{Z}^{d},\mathbb{E}^{d})$. Throughout this section,
$G=(V,E)$ is a finite subgraph of $\mathbb{G}$.

### Proof of Item 1

Let us define the critical point $\beta_{c}=\beta_{c}(q,\bm{\widehat{h}})$ by

(9) $\displaystyle\beta_{c}=\inf\big{\\{}\beta\geqslant
0:\theta(\beta,q,\bm{\widehat{h}})>0\big{\\}},$

where $\theta$ is the percolation-probability defined in (4). Fix $x\in V$.
For each $n\geqslant 1$, let $B(x,n)=\big{\\{}y\in V:\|x-y\|\leqslant
n\big{\\}}$ be a box of radius $n$ with centre at $x$ and $\partial
B(x,n)=\big{\\{}y\in\mathbb{Z}^{d}:\|x-y\|=n\big{\\}}$ its surface. Since
$\big{\\{}x\longleftrightarrow\partial B(x,n)\big{\\}}$ is an increasing
event, by combining Lemma 3.7 and Corollary 3.11, it follows that

$\displaystyle\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{0}}}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\leqslant\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\leqslant\mathbb{P}_{G;\beta}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}.$

Letting $G\uparrow\mathbb{G}$, from Theorem 3.3-Item (a) we have

(10) $\displaystyle\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{0}}}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\leqslant\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\leqslant\mathbb{P}_{\mathbb{G};\beta}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]},$

where $\mathbb{P}_{\mathbb{G};\beta}$ denotes the infinite-volume Bernoulli
measure. Since $\\{x\longleftrightarrow\partial
B(x,n)\\}\downarrow\\{|C_{x}|=\infty\\}$ as $n\to\infty$, it follows from the
continuity of the measure that

$\displaystyle\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{0}}}\big{(}|C_{x}|=\infty\big{)}\leqslant\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|=\infty\big{)}\leqslant\mathbb{P}_{\mathbb{G};\beta}\big{(}|C_{x}|=\infty\big{)},\quad
x\in\mathbb{Z}^{d}.$

Then, by using the definition of function $\theta$,

(11)
$\displaystyle\theta(\beta,q,\bm{\widehat{0}})\leqslant\theta(\beta,q,\bm{\widehat{h}})\leqslant\theta(\beta,1,\bm{\widehat{0}}),\quad\beta\in[0,\infty].$

By definition of $\beta_{c}$, this implies that

(12)
$\displaystyle\beta_{c}(q,\bm{\widehat{0}})\geqslant\beta_{c}(q,\bm{\widehat{h}})\geqslant\beta_{c}(1,\bm{\widehat{0}}).$

Since
$0<1-\exp\big{[}{-q\beta_{c}(1,\bm{\widehat{0}})\,J}\big{]},1-\exp\big{[}{-q\beta_{c}(q,\bm{\widehat{0}})\,J}\big{]}<1$
for $d\geqslant 2$ (see [27, 28]), we deduce the important fact that

$0<\beta_{c}(q,\bm{\widehat{h}})<\infty,\quad q\in\\{2,3,\ldots\\}.$

That is, the critical point $\beta_{c}$ is non-trivial. On the other hand, it
is known that $\theta(\beta,q,\bm{\widehat{0}})>0$ when
$\beta>\beta_{c}(q,\bm{\widehat{0}})$, and that
$\theta(\beta,1,\bm{\widehat{0}})=0$ when
$\beta<\beta_{c}(\bm{\widehat{0}},1).$ Using this in (11) we have

$\theta(\beta,q,\bm{\widehat{h}})\left\\{\begin{array}[]{lllll}=0&\text{if}\
\beta<\beta_{c}(1,\bm{\widehat{0}}),\\\\[2.84544pt] >0&\text{if}\
\beta>\beta_{c}(q,\bm{\widehat{0}}).\end{array}\right.$

Since $\theta$ is an increasing function in $\beta$, see Proposition 3.9-Item
(b), the proof follows.

∎

######  Remark 1.

Consider the same notation as the proof of Item 1). For $\beta<\beta_{0}$,
where $\beta_{0}>0$ is a certain real number less than $\beta_{c}$, given in
(9), there exists $\psi(\beta)>0$ such that

(13) $\displaystyle\sup_{x\in\mathbb{Z}^{d}}\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\leqslant{\exp}\big{[}-n\psi(\beta)\big{]}\quad\text{for all}\
n.$

In other words, we have the exponential tail decay of the radius of an open
cluster for sufficiently small parameter values. Indeed, by (10), we have

(14) $\displaystyle\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\leqslant\mathbb{P}_{\mathbb{G};\beta}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]},\quad x\in\mathbb{Z}^{d}.$

Moreover, for any $\beta<\beta_{c}(1,\bm{\widehat{0}})$ it is known that there
exists $\psi(\beta)>0$ such that, see, e.g., [28, 39], or Corollary 9.38 in
[28],

(15)
$\displaystyle\mathbb{P}_{\mathbb{G};\beta}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\leqslant{\exp}\big{[}-n\psi(\beta)\big{]}\quad\text{for all}\
n.$

By combining (14) and (15), the inequality in (13) follows by taking
$\beta_{0}=\beta_{c}(1,\bm{\widehat{0}})$.

######  Remark 2.

In the random-cluster model a percolation transition is guaranteed, because
$\phi^{\rm w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(N_{\infty}=1)=0$ when
$\beta<\beta_{c},$ and $\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(N_{\infty}=1)=1$ when
$\beta>\beta_{c}$. Indeed, from Theorem 1-Item 1), for any $\beta>\beta_{c}$,

$0<\theta(\beta,q,\bm{\widehat{h}})=\sup_{x\in\mathbb{Z}^{d}}\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|=\infty\big{)}\leqslant\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(N_{\infty}\geqslant 1).$

Since $\\{N_{\infty}\geqslant 1\\}$ is a tail event and $\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}$ is an extremal measure (see Theorem
3.3-Item (b)), it follows from the uniqueness of the infinite cluster (see
Theorem 3.4) that $\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(N_{\infty}=1)=1$. On the other hand,
if $\beta<\beta_{c}$, by Theorem 1-Item 1), we have $\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(|C_{x}|=\infty)=0$ for all
$x\in\mathbb{Z}^{d}$. Again, from the uniqueness of the infinite cluster,

$\displaystyle 0=\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(N_{\infty}>0)=\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(N_{\infty}=1).$

Finally, since the infinite-volume measure is unique when $\beta<\beta_{c}$,
that is, $\phi^{\rm f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}=\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}$ (see Theorem 11-Item (ii) in [11]),
we have $\phi^{\rm w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(N_{\infty}=1)=0.$

### Proof of Item 2

We follow a similar argument as in reference [27] p. 46 and the same notation
as the proof of Item 1). Let $M_{x}$ be the random variable defined by
$M_{x}=\max\big{\\{}n:\\{x\longleftrightarrow\partial B(x,n)\\}\
\text{occurs}\big{\\}}$ for each $x\in\mathbb{Z}^{d}$. If $\beta<\beta_{c}$,
by Remark 2 we have $\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(M_{x}<\infty)=1$, that is, the
collection $\\{M_{x}=n\\}_{n\geqslant 1}$ is a partition of the sample space
$\\{0,1\\}^{\mathbb{E}^{d}}$. This gives

$\displaystyle\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|\big{)}$
$\displaystyle\leqslant\sum_{n=1}^{\infty}\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|\big{|}M_{x}=n\big{)}\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(M_{x}=n)$
$\displaystyle\leqslant\sum_{n=1}^{\infty}\big{|}B(x,n)\big{|}\,\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\quad\text{for all}\ x\in\mathbb{Z}^{d}.$

Since $|B(x;n)|\leqslant\pi(d)(n+1)^{d}$ for some constant $\pi(d)$, from
Remark 1 we have, for $\beta<\beta_{0}$,

$\displaystyle\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|\big{)}\leqslant\pi(d)\sum_{n=1}^{\infty}(n+1)^{d}\,{\exp}\big{[}-n\psi(\beta)\big{]}<\infty\quad\text{for
all}\ x\in\mathbb{Z}^{d}.$

Taking supremum on $x$ in the above inequality, by definition (4) of the
susceptibility function $\chi$, we have that $\chi$ is finite when
$\beta<\beta_{0}$. On the other hand, note that

$\displaystyle\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|\big{)}$
$\displaystyle=\infty\bm{\cdot}\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|=\infty\big{)}+\sum_{n=1}^{\infty}n\,\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|=n\big{)}$
$\displaystyle\geqslant\infty\bm{\cdot}\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{(}|C_{x}|=\infty\big{)}\quad\text{for
all}\ x\in\mathbb{Z}^{d}.$

Taking supremum on $x$ in the above inequality, by definition (4) of $\theta$
and $\chi$, we have

$\displaystyle\chi(\beta,q,\bm{\widehat{h}})\geqslant\infty\bm{\cdot}\theta(\beta,q,\bm{\widehat{h}}).$

By Theorem 1-Item 1), $\theta(\beta,q,\bm{\widehat{h}})>0$ when
$\beta>\beta_{c}$. Then, it follows that
$\chi(\beta,q,\bm{\widehat{h}})=\infty$ when $\beta>\beta_{c}$. Thus the proof
of the second item is complete.

∎

######  Remark 3.

There exists a critical point $0<\pi_{c}=\pi_{c}(q,\bm{\widehat{h}})<\infty$
such that $\chi(\beta,q,\bm{\widehat{h}})<\infty$ if $\beta<\pi_{c},$ and
$\chi(\beta,q,\bm{\widehat{h}})=\infty$ if $\beta>\pi_{c}.$ In other words,
the susceptibility function $\chi$ experiences a phase transition phenomenon.
Indeed, let us define

$\displaystyle\pi_{c}=\sup\big{\\{}\beta\geqslant
0:\chi(\beta,q,\bm{\widehat{h}})<\infty\big{\\}},$

where $\chi$ is as in (4). Note that $\pi_{c}\leqslant\beta_{c}$, where
$\beta_{c}$ is the critical probability given in (9). Since the event
$\\{x\longleftrightarrow y\\}$ is increasing, by combining Lemma 3.7 and
Corollary 3.11, in the thermodynamic limit, we have

$\displaystyle\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{0}}}(x\longleftrightarrow
y)\leqslant\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow
y)\leqslant\mathbb{P}_{\mathbb{G};\beta}(x\longleftrightarrow y)\quad\text{for
all}\ x,y\in\mathbb{Z}^{d}.$

Combining this with the identity

(16)
$\displaystyle\chi(\beta,q,\bm{\widehat{h}})=\sup_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}}\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow y),$

we obtain
$\chi(\beta,q,\bm{\widehat{0}})\leqslant\chi(\beta,q,\bm{\widehat{h}})\leqslant\chi(\beta,1,\bm{\widehat{0}}).$
Consequently, by definition of $\pi_{c}$, we have

(17)
$\displaystyle\pi_{c}(1,\bm{\widehat{0}})\leqslant\pi_{c}(q,\bm{\widehat{h}})\leqslant\pi_{c}(q,\bm{\widehat{0}}).$

It is well-known that
$\pi_{c}(1,\bm{\widehat{0}})=\beta_{c}(1,\bm{\widehat{0}})$ (see [2, 34, 35])
and that $\pi_{c}(q,\bm{\widehat{0}})\leqslant\beta_{c}(q,\bm{\widehat{0}})$.
Since
$0<1-\exp\big{[}{-q\beta_{c}(1,\bm{\widehat{0}})\,J}\big{]},1-\exp\big{[}{-q\beta_{c}(q,\bm{\widehat{0}})\,J}\big{]}<1$
for $d\geqslant 2$ (see [27, 28]), we have that $\pi_{c}(q,\bm{\widehat{h}})$
is non-trivial for all $q\in\\{2,3,\ldots\\}.$ On the other hand, by Theorem
1-Item 2), $\chi(\beta,q,\bm{\widehat{h}})=\infty$ when $\beta>\beta_{c}$, and
$\chi(\beta,q,\bm{\widehat{h}})<\infty$ when $\beta<\beta_{0}$. Since $\chi$
is an increasing function in $\beta$, see Proposition 3.9-Item (b), the remark
follows.

### Proof of Item 3

From Lemma 3.12 with $f=\mathds{1}_{\\{x\longleftrightarrow y\\}}$, in the
thermodynamic limit, we have

$\displaystyle\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow y)\leqslant
C(\beta,\bm{\widehat{h}})\,\phi^{\rm
f}_{\mathbb{G};\beta,q,\bm{\widehat{0}}}(x\longleftrightarrow y)\quad\text{for
all}\ x,y\in\mathbb{Z}^{d},$

whenever $\bm{\widehat{h}}\in\ell^{1}(\mathbb{Z}^{d})$ has positive terms. It
is known that, for $\beta<\beta_{c}(q,\bm{\widehat{0}})$ and
$q\in\\{2,3,\ldots\\}$, there exists $\gamma(\beta,q)>0$ such that (see main
theorem in [18]),

$\phi^{\rm f}_{\mathbb{G};\beta,q,\bm{\widehat{0}}}(x\longleftrightarrow
y)\leqslant{\exp}\big{[}{-\|x-y\|\gamma(\beta,q)}\big{]}.$

And thereby, for $\beta<\beta_{c}(q,\bm{\widehat{0}})$,

(18) $\displaystyle\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow y)\leqslant
C(\beta,\bm{\widehat{h}})\,{\exp}\big{[}{-\|x-y\|\gamma(\beta,q)}\big{]}\quad\text{for
all}\ x,y\in\mathbb{Z}^{d}.$

In what follows we claim that
$\beta_{c}(q,\bm{\widehat{0}})=\beta_{c}(q,\bm{\widehat{h}})$. Indeed, by Item
(12), $\beta_{c}(q,\bm{\widehat{0}})\geqslant\beta_{c}(q,\bm{\widehat{h}}).$
Now, suppose that
$\beta_{c}(q,\bm{\widehat{0}})>\beta_{c}(q,\bm{\widehat{h}})$. From Theorem
1-Item 2),

(19) $\displaystyle\chi(\beta,q,\bm{\widehat{h}})=\infty\quad\text{when}\
\beta_{c}(q,\bm{\widehat{h}})<\beta<\beta_{c}(q,\bm{\widehat{0}}).$

From (18), for $\beta<\beta_{c}(q,\bm{\widehat{0}})$ and for all
$y\in\mathbb{Z}^{d}$,

$\displaystyle\sum_{y\in\mathbb{Z}^{d}}\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow y)\leqslant
C(\beta,\bm{\widehat{h}})\,\sum_{n=1}^{\infty}{\exp}\big{[}{-n\gamma(\beta,q)}\big{]}\Bigg{(}\sum_{y:\|x-y\|=n}1\Bigg{)}<\infty.$

By combining this with the identity (16), we have
$\chi(\beta,q,\bm{\widehat{h}})<\infty$ when
$\beta<\beta_{c}(q,\bm{\widehat{0}})$, but this contradicts the statement in
(19). Hence the claimed follows.

Therefore, the exponential decay (18) is satisfied for all
$\beta<\beta_{c}(q,\bm{\widehat{h}})$. This completes the proof of the third
item.

∎

######  Remark 4.

Considering the same notation as the proof of Item 1), notice that the third
item of Theorem 1 could be replaced by the following statement. In the case
that $\bm{\widehat{h}}\in\ell^{1}(\mathbb{Z}^{d})$ has positive terms, for all
$\beta>0$ there exist $C(\beta,\bm{\widehat{h}}),\gamma(\beta,q)>0$, such that

$\displaystyle\sup_{x\in\mathbb{Z}^{d}}\phi^{\rm
w}_{\mathbb{G};\beta,q,\bm{\widehat{h}}}\big{[}x\longleftrightarrow\partial
B(x,n)\big{]}\leqslant
C(\beta,\bm{\widehat{h}})\,{\exp}\big{[}-n\gamma(\beta,q)\big{]}\quad\text{for
all}\ n,$

whenever $\beta<\beta_{c}$.

######  Remark 5.

Let $x,z\in V$ be distinct vertices. A subset $W$ of $V$ is said to separate
$x$ and $z$ if $x,z\notin W$ and every path from $x$ to $z$ contains some
vertex of $W$. Let $q=2$ and let $\bm{\widehat{h}}\in\ell^{1}(\mathbb{Z}^{d})$
be a summable positive external field. Assume that $x,z\in V$ are distinct
vertices, and that $W$ separate $x$ and $z$. For $\beta\geqslant 0$ there
exists $C(\beta,\bm{\widehat{h}})>0$ such that

(20) $\displaystyle\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow z)\leqslant
C(\beta,\bm{\widehat{h}})\sum_{y\in W}\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow y)\,\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}(y\longleftrightarrow z).$

The above correlation inequality is known as the (modified) Simon-Lieb
inequality. Indeed, since the event $\\{x\longleftrightarrow y\\}$ is monotone
increasing, Corollary 3.11 gives

(21) $\displaystyle\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{0}}}(x\longleftrightarrow y)\leqslant\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow y)\quad\text{for all}\
x,y\in V.$

The Simon-Lieb inequality [28, 29, 31, 39] for the measure
$\phi_{G;\beta,q,\bm{\widehat{0}}}$, with $q=2$, gives

$\displaystyle\phi^{\rm f}_{G;\beta,q,\bm{\widehat{0}}}(x\longleftrightarrow
z)$ $\displaystyle\leqslant\sum_{y\in W}\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{0}}}(x\longleftrightarrow y)\,\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{0}}}(y\longleftrightarrow z)$
$\displaystyle\stackrel{{\scriptstyle\eqref{ineq-
fund-1}}}{{\leqslant}}\sum_{y\in W}\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}(x\longleftrightarrow y)\,\phi^{\rm
f}_{G;\beta,q,\bm{\widehat{h}}}(y\longleftrightarrow z).$

By combining this with Lemma 3.12, the validity of the inequality in (20)
follows.

######  Remark 6.

Since $C(\beta,\bm{\widehat{h}})\geqslant 1$ and $\phi^{\rm
f}_{\mathbb{G},\beta,q,\bm{\widehat{h}}}$ is a non-translation-invariant
measure with $q=2$ and $\bm{\widehat{h}}\in\ell^{1}(\mathbb{Z}^{d})$ has
positive terms, note that we cannot use the Simon-Lieb inequality in (20), as
in the translationally-invariant case, to prove the exponential decay of the
connectivity function for two points.

### Acknowledgments

We would like to thank L. Cioletti for many valuable comments and careful
reading of this manuscript. This study was financed in part by the Coordenação
de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance
Code 001.

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# How well do Large Language Models perform in Arithmetic tasks?

Zheng Yuan1 Hongyi Yuan12 Chuanqi Tan1 Wei Wang1 Songfang Huang1
1Alibaba Group 2Tsinghua University
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

Large language models have emerged abilities including chain-of-thought to
answer math word problems step by step Wei et al. (2022b). Solving math word
problems not only requires abilities to disassemble problems via chain-of-
thought but also needs to calculate arithmetic expressions correctly for each
step. To the best of our knowledge, there is no work to focus on evaluating
the arithmetic ability of large language models. In this work, we propose an
arithmetic dataset MATH 401 to test latest large language models including
GPT-4, ChatGPT, InstrctGPT, Galactica, and LLaMA with various arithmetic
expressions and provide a detailed analysis of the ability of large language
models. MATH 401 and evaluation codes are released at
https://github.com/GanjinZero/math401-llm. 111This project is working in
progress.

##  1 Introduction

Emergent abilities show in sufficiently large language models (LLMs) Wei et
al. (2022a) like chain-of-thought reasoning (COT) Wei et al. (2022b). Chain-
of-thought reasoning requires LLMs to solve a question by thinking questions
step by step which performs well in school math word problems Wei et al.
(2022b); Kojima et al. (2022). Recent LLMs are further fine-tuned with
instruction tuning Sanh et al. (2021); Chung et al. (2022); Ouyang et al.
(2022) which demonstrates improved COT ability compared to only self-
supervised pre-training. To solve a math word problem, COT disassembles the
problem into simple steps. For each step, LLMs have to compute correctly based
on arithmetic expressions. Thus, evaluating the arithmetic ability of LLMs is
necessary since it is the upper bound of LLMs’ ability for solving math word
problems.

To this end, we propose an arithmetic dataset named MATH 401. Different
difficulties are contained in this dataset including addition ($+$),
subtraction ($-$), multiplication ($\times$), division ($\div$),
exponentiation ($\wedge$), trigonometry functions ($\sin,\cos,\tan$), and
logarithm functions ($\log,\ln$) of integers, decimals, and irrational numbers
($\pi,e$). Long arithmetic expressions with brackets are also included which
are common in complex math word problems. Results in Table 1 show detailed
evaluations on OpenAI’s GPTs including GPT-4 OpenAI (2023),
ChatGPT222https://openai.com/blog/introducing-chatgpt-and-whisper-apis,
GPT-3.5 Ouyang et al. (2022) and other open-sourced LLMs. We find that GPT-4
and ChatGPT outperform other models by a large margin in all kinds of
arithmetic abilities. InstructGPT Ouyang et al. (2022) and Galactica Taylor et
al. (2022) do have some arithmetic abilities. We analyze factors affecting
LLMs’ arithmetic ability systematically including tokenization (Section 4.2),
pre-training (Section 4.3), prompts (Section 4.4), interpolation and
extrapolation (Section 4.5), scaling laws (Section 4.6), COT (Section 4.7),
and ICL (Section 4.8).

Model | Size | E | $+-$ | $\times$ | $\div$ | $\wedge$ | Tri | $\log$ | Dec | Neg | Irr | Big | Long | Easy | Hard | All
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
GPT-4 | ? | $\checkmark$ | 99 | 67 | 100 | 50 | 68 | 76 | 67 | 67 | 100 | 48 | 96 | 100 | 67 | 84
ChatGPT | ? | $\checkmark$ | 97 | 65 | 80 | 50 | 44 | 56 | 67 | 67 | 64 | 40 | 68 | 100 | 49 | 74
InstructGPT | 175B | $\times$ | 83 | 59 | 80 | 36 | 8 | 16 | 64 | 64 | 36 | 4 | 24 | 92 | 22 | 57
CodeX | 175B | $\checkmark$ | 36 | 27 | 8 | 10 | 8 | 0 | 25 | 25 | 12 | 0 | 0 | 40 | 4 | 22
Galactica | 120B | $\checkmark$ | 69 | 43 | 24 | 44 | 16 | 0 | 57 | 57 | 28 | 0 | 24 | 78 | 12 | 45
LLaMA | 65B | $\checkmark$ | 44 | 35 | 8 | 22 | 8 | 0 | 41 | 41 | 20 | 0 | 4 | 52 | 5 | 28
OPT | 175B | $\checkmark$ | 33 | 35 | 4 | 12 | 0 | 4 | 25 | 25 | 8 | 0 | 0 | 41 | 2 | 22
GPT-Neox | 20B | $\checkmark$ | 51 | 48 | 4 | 40 | 4 | 0 | 43 | 43 | 20 | 0 | 8 | 66 | 4 | 35
GLM | 130B | $\checkmark$ | 39 | 31 | 8 | 22 | 0 | 0 | 29 | 29 | 24 | 0 | 8 | 46 | 5 | 26
BloomZ | 176B | $\times$ | 23 | 37 | 12 | 30 | 8 | 0 | 43 | 43 | 20 | 0 | 8 | 39 | 6 | 22
Bloom | 176B | $\times$ | 21 | 37 | 12 | 30 | 0 | 0 | 37 | 37 | 16 | 0 | 0 | 37 | 4 | 20
T0++ | 11B | $\times$ | 6 | 3 | 0 | 6 | 8 | 0 | 3 | 3 | 4 | 0 | 0 | 7 | 2 | 4
Flan-T5 | 11B | $\times$ | 1 | 13 | 4 | 0 | 0 | 0 | 11 | 11 | 8 | 0 | 0 | 6 | 2 | 4
Table 1: Arithmetic ability for LLMs measured by accuracy, we only list models
with largest parameter counts. E = Euler, Dec = Decimal, Neg = Negative, Irr =
Irrational, Big = Big Numbers, Long = Long Expressions.

One may say that the ability to solve arithmetic tasks is not necessary for a
large language model. LLMs can use the calculator API when they need to decode
an answer Schick et al. (2023). Arithmetic ability evaluation can be a gauge
for general intelligence since mastering arithmetic serves as a fundamental
requirement for performing intricate mathematical tasks including symbolic
math reasoning Noorbakhsh et al. (2021); Gaur and Saunshi (2022) and automatic
theorem proofing Polu and Sutskever (2020); Wu et al. (2022).

##  2 Related Works

#### Evaluate Math Ability of LLMs

To show the math reasoning ability of LLMs, Wang and Komatsuzaki (2021); Chung
et al. (2022); Thoppilan et al. (2022) evaluate their models on various math
word problems benchmark Saxton et al. (2019); Hendrycks et al. (2021); Cobbe
et al. (2021); Shi et al. (2022). For newly released LLM ChatGPT, Shakarian et
al. (2023); Frieder et al. (2023) evaluate its mathematical ability
independently. To notice, our paper evaluates ChatGPT using gpt-3.5-turbo-0301
version and GPT-4 using chat UI on March 16th which may have different
performances compared to their reported results and future analysis.

#### Evaluate Arithmetic Ability of LLMs

Nogueira et al. (2021); Wang et al. (2021) evaluate pretrained language models
on simple arithmetic expressions including addition ($+$) and subtraction
($-$). Muffo et al. (2022) have further tested the multiplication ($\times$)
of language models. They found tokenization Nogueira et al. (2021); Kim et al.
(2021) and token frequency Razeghi et al. (2022) are two important factors for
language model arithmetic ability. Compared to previous work, we focus on
evaluating Large LMs (with instruction fine-tuning) on comprehensive
arithmetic abilities with different types of operators and numbers.

##  3 Evaluation Settings

###  3.1 Arithmetic Expression Settings

We construct 401 arithmetic expressions to test large language models which
include Euler equation ($e^{i\pi}+1=0$) as group 0 and 25 problems each for
group 1$\sim$16\. If not otherwise mentioned, used numbers are positive
integers.

  * •

Euler Equation.

  * •

Add & Subtract of two integers within 10.

  * •

Add & Subtract of two integers within 100.

  * •

Add & Subtract of two integers within 1,000.

  * •

Add & Subtract of two integers within 1,000,000,000,000.

  * •

Add & Subtract of two integers within -10$\sim$10.

  * •

Add & Subtract of two decimal numbers within -100$\sim$100.

  * •

Multiply two integers within 100.

  * •

Multiply two decimal numbers within 10.

  * •

Multiply two integers within 100,000.

  * •

Division of two integers within 100.

  * •

Exponentiation of with integer base within 10 and integer exponent within
2$\sim$4.

  * •

Exponentiation of with a decimal number within 10 as the base and a decimal
number within 2$\sim$4 as the exponent.

  * •

Add, Subtract & Multiply with one integer within 10 and a common irrational
number (i.e. $e$ or $\pi$).

  * •

Long arithmetic expressions with brackets, involved integers are all within
100 and operators contain add, subtract, multiply, and division.

  * •

Trigonometry functions including $\sin$, $\cos$, and $\tan$. Inputs can be in
the format of degrees and radians ($\pi$ can also appear in the inputs).

  * •

Logarithm of integers within 1000 of different bases: $2,e,10$.

These groups cover mathematical operators used in elementary mathematics. We
consider groups 1,2,3,5,6,7,8,11 as Easy queries and all others as Hard
queries. We calculate the results of all arithmetic expressions using built-in
functions of Python and round to four decimal places. Examples of expressions
are listed in Appendix A.

###  3.2 Metrics

Since LLMs can decode arbitrary contents (which may contain their step-by-step
calculation steps), we first ignore decoded numbers in parentheses and
preserve the last number decoded by LLMs. If the decoded number is a fraction,
we will convert it to decimal for evaluation except for group 10 which
requires calculating division. To measure the arithmetic ability of LLMs, we
use the following metrics to measure their outputs.

#### Accuracy

If the difference between the decoded number and the target number is less
than $1e-3$, we consider it a correct prediction. Accuracy is calculated based
on correct prediction counts.

#### Relative error

We denote decoded number is $\hat{y}$ and target is $y$. We calculate relative
error by:

$RE=\min(10,\frac{\|\hat{y}-y\|}{\max(\|y\|,1)})$ (1)

If LLM does not decode any number, we consider $RE=10$. We truncate the
relative error to 10 to prevent that one big mistake dominate the average
relative error.

#### Non-number ratio

If decoded content does not contain any numbers, we consider it a failure. We
calculate the non-number ratio based on it.

###  3.3 Evaluation Details

We test GPT-4 by their official chat
UI333https://chat.openai.com/chat?model=gpt-4. Since GPT-4 has limited request
counts, we only query GPT-4 with groups that ChatGPT cannot answer correctly.
We test GPT-3.5 (including davinci (CodeX, InstructGPT) and turbo (ChatGPT)
series models) Ouyang et al. (2022); Chen et al. (2021) via OpenAI APIs. We
also test following open-sourced LLMs including Galactica Taylor et al.
(2022), GPT from EleutherAI Wang and Komatsuzaki (2021); Black et al. (2022),
LLaMA Touvron et al. (2023), OPT (with instruction learning) Zhang et al.
(2022), Bloom (with instruction learning) Scao et al. (2022); Muennighoff et
al. (2022), T0++ Sanh et al. (2021), GLM Zeng et al. (2022) and Flan-T5 Chung
et al. (2022). We also test the smaller versions of the above models.

We test following prompts: $\emptyset$ (i.e. no prompt), “Calculate:”, “$”,
“$$”, and “\begin{equation}”. The latest three prompts are inspired by that
LLMs may be pretrained with LaTeX sources. We provide three versions of input
formats: math texts ($\pi$), plain texts (pi), LaTeX texts (\pi). When we use
LaTeX-related prompts, we provide the model with LaTeX texts. When we use
other prompts, we will provide math texts if their tokenizers can encode them.
Otherwise, we will provide plain text. For ChatGPT (gpt-3.5-turbo-0301), we
test different system-level prompts as instructions: $\emptyset$ (i.e. no
prompt), “You are an accurate calculator.”, and “You are an accurate
calculator, please calculate provided equation to four decimal places.”. For
GPT-4, we only test prompt “You are an accurate calculator, please calculate
provided equation to four decimal places.”.

We use default decode settings for OpenAI APIs, and we use greedy decoding for
all other LLMs.

Model | Prompt | Acc $\uparrow$ | RE $\downarrow$ | NNR $\downarrow$
---|---|---|---|---
gpt-4 | Cal*4 | 83.54 | 0.07 | 0.00
gpt-3.5-turbo-0301 | Cal* | 75.06 | 0.14 | 0.50
text-davinci-003 | Cal | 56.61 | 0.76 | 2.99
code-davinci-002 | Eqa | 21.7 | 2.39 | 11.47
galactica-120b | Eqa | 45.14 | 1.30 | 3.99
galactica-30b | Eqa | 45.14 | 0.69 | 1.75
llama-65b | Eqa | 28.43 | 1.61 | 4.74
opt-175b | Cal | 21.70 | 3.18 | 21.70
gpt-neox-20b | Eqa | 35.41 | 1.19 | 4.49
glm-130b | $ | 25.94 | 1.27 | 2.74
bloomz-176b | $$ | 22.44 | 1.50 | 4.74
bloom-176b | $ | 20.20 | 2.60 | 18.45
T0++-11b | Cal | 4.24 | 3.34 | 9.48
flan-t5-xxl-11b | Eqa | 3.74 | 5.78 | 43.89
flan-t5-xl-3b | $ | 7.48 | 3.34 | 25.19
Table 2: Evaluation on MATH 401 with different LLMs. Prompts are selected via best accuracy. Cal means “Calculate:” and Eqa means “\begin{equation}”. * means providing an additional system-level message. Model | Prompt | Acc $\uparrow$ | RE $\downarrow$ | NNR $\downarrow$
---|---|---|---|---
gpt-4 | Cal*4 | 83.54 | 0.07 | 0.00
gpt-3.5-turbo-0301 | Cal* | 75.06 | 0.14 | 0.50
text-davinci-003 | Cal | 56.61 | 0.76 | 2.99
text-davinci-002 | Cal | 42.89 | 2.13 | 15.96
text-curie-001 | Cal | 11.47 | 1.92 | 6.48
text-babbage-001 | Eqa | 5.24 | 2.59 | 5.74
code-davinci-002 | Eqa | 21.70 | 2.39 | 11.47
galactica-120b | Eqa | 45.14 | 1.30 | 3.99
galactica-30b | Eqa | 45.14 | 0.69 | 1.75
galactica-6.7b | Cal | 34.41 | 2.61 | 8.73
llama-65b | Eqa | 28.43 | 1.61 | 4.74
llama-30b | Eqa | 30.17 | 1.72 | 3.74
llama-13b | $ | 27.68 | 2.40 | 9.73
llama-7b | $$ | 21.95 | 2.11 | 7.48
opt-175b | Cal | 21.70 | 3.18 | 21.70
opt-66b | $\emptyset$ | 20.70 | 2.66 | 18.70
opt-iml-max-30b | Cal | 17.46 | 1.52 | 6.23
opt-30b | $\emptyset$ | 15.96 | 2.28 | 11.22
opt-13b | $\emptyset$ | 15.21 | 2.19 | 10.97
opt-6.7b | Cal | 14.46 | 1.46 | 4.24
gpt-neox-20b | Eqa | 35.41 | 1.19 | 4.49
gpt-j-6b | Cal | 27.18 | 1.55 | 8.98
bloomz-176b | $$ | 22.44 | 1.50 | 4.74
bloom-176b | $ | 20.2 | 2.60 | 18.45
bloomz-7b1 | $ | 12.72 | 2.56 | 15.46
bloom-7b1 | Cal | 7.23 | 2.41 | 6.48
bloomz-3b | $$ | 7.98 | 2.63 | 12.47
bloom-3b | Cal | 4.24 | 2.41 | 8.73
bloomz-1b7 | Eqa | 4.74 | 4.28 | 31.17
bloom-1b7 | Cal | 5.24 | 2.54 | 11.22
T0++-11b | Cal | 4.24 | 3.34 | 9.48
glm-130b | $ | 25.94 | 1.27 | 2.74
glm-10b | Cal | 14.96 | 2.30 | 3.74
flan-t5-xxl-11b | Eqa | 3.74 | 5.78 | 43.89
flan-t5-xl-3b | $ | 7.48 | 3.34 | 25.19
flan-t5-large-780m | Cal | 3.74 | 2.31 | 2.49
flan-t5-base-250m | Eqa | 2.49 | 3.18 | 14.21
Table 3: Full evaluation on MATH 401 with different LLMs. Prompts are selected
via best accuracy.

##  4 Results and Analysis

###  4.1 Results

#### Overall Results

Table 1, 2, and 3 show results of different LLMs on MATH 401. We find GPT-4
and ChatGPT outperform all other models by a large margin444OpenAI states they
improve the math of ChatGPT since version Jan 30, and we cannot evaluate any
previous version.. GPT-4 surpasses ChatGPT with accuracy of 10 points and
reduce relative error half. InstructGPT performs third measured by accuracy
and Galactica-30B performs third measured by relative error. Compared to
models proposed before InstructGPT (text-davinci-003), GPT-series applies
Reinforcement Learning from Human Feedback (RLHF) which may enhance their
arithmetic ability significantly. Galactica is pre-trained with massive LaTeX
source codes which could be the reason why Galactica performs well in
arithmetics.

#### Grouped Results

To clearly understand the arithmetic ability of LLMs, we show grouped accuracy
in Table 1. GPT-4 obtains first places and ChatGPT obtains second places for
all groups. Most LLMs are only capable of doing addition and subtraction and
have some ability for multiplication. Division, exponentiation, trigonometry
functions, and logarithm functions are hard for most LLMs. LLMs have some
abilities dealing with decimal, negative, and irrational numbers. Only GPT-4
and ChatGPT have the ability to deal with big numbers ($>1e12$) and complex
long queries which proves their generalization and reasoning abilities. GPT-4
shows extremely good ability in long arithmetic expressions.

#### When will ChatGPT fail?

Though ChatGPT obtains such a good performance, we will check when ChatGPT
fails to answer. For multiplication ($\times$), ChatGPT passes all queries in
Group 7 and 8 and get wrong answers for all queries in Group 9. An example is
ChatGPT predicts $71786\times
21638=1,55{\color[rgb]{1,0,0}1},{\color[rgb]{1,0,0}4}0{\color[rgb]{1,0,0}2,0}68$,
while the true answer is $1,553,305,468$. ChatGPT gives a very close
estimation with the correct head and tail, which proves that ChatGPT does not
use a calculator API for math calculation.

For division in Group 11, ChatGPT sometimes gives correct answers to two
decimal places which will be considered incorrect in our metric. We can see in
Table 5, requiring ChatGPT to output four decimal places will improve its
accuracy in multiplication and division.

For exponentiation ($\wedge$), ChatGPT correctly answers all queries in Group
10 which contain only integers as bases. It is too hard for any language model
(even ChatGPT) correctly estimate the exponentiation of a decimal number as
the base and a decimal number as the exponent. It seems that ChatGPT treats
$**$ as multiplication sometimes, for example, ChatGPT estimates $5.5507$ $**$
$2.0434={\color[rgb]{1,0,0}10.31554}$ which is close to $5.5507\times
2.0434=11.3423$ and far from answer $33.1895$.

For calculating trigonometry functions, ChatGPT understands degrees and
radians correctly and generates exact answers for special inputs like
$cos(-210^{\circ})=-\frac{\sqrt{3}}{2}$ (we omit explanation generated by
ChatGPT here). However, ChatGPT may generate wrong explanations which mislead
itself. An example is: “We know that the sine function is periodic with a
period of $2\pi$, which means that $\sin(x+2\pi)=\sin(x)$ for any value of
$x$. Therefore, we can subtract multiples of $2\pi$ from $-3.75\pi$ until we
get a value between 0 and $2\pi$:
$-3.75\pi=-3\pi-0.75\pi=-9.42477-2.35619=-11.78096$. Adding 2$\pi$, we get:
$-11.78096+2\pi$ = -9.42477 etc.” Any mistake in explanations may result in a
wrong answer.

For logarithm functions, we find that ChatGPT is capable of using change of
base formula and predicting answers within two decimal places.

For long expressions, ChatGPT can understand the operators’ priorities.
ChatGPT sometimes generates answers step by step and sometimes generates
answers directly. It is very likely to generate wrong answers when it decodes
answers directly.

#### What about GPT-4?

For big number multiplication ($\times$) in group 9, GPT-4 also fails in all
cases with similar problems occurring in ChatGPT.

For exponentiation ($\wedge$), GPT-4 will not consider $**$ as $\times$
anymore and give better estimations.

For calculating expressions with irrational numbers, GPT-4 will consider $e$
as natural logarithm correctly.

For logarithm functions, GPT-4 calculates logarithm base $e$ and 10 by “using
a calculator” (this is a message generated by GPT-4). GPT-4 calculates
logarithm base 2 by change of base formula and generates approximate results.

For long equations, GPT-4 solves all equations step by step and obtains a much
higher accuracy.

We compare and summarize how GPT-4 outperforms ChatGPT here:

  * •

Better division ability.

  * •

Better trigonometry ability.

  * •

Understand irrational numbers properly.

  * •

Always calculate long expressions step by step.

###  4.2 Tokenization

Arithmetic expressions have special tokens including
$\pi,\times,\div,{}^{\circ}$ which are not within T5 series models (i.e. T0++
and Flan-T5). T0++-11B (Acc 4.24 and RE 3.34) and Flan-T5-xxl-11B (Acc 3.74
and RE 5.78) perform badly on arithmetic tasks compared to other similar-size
models: Opt-13B (Acc 15.21 and RE 2.19) and LLaMA-13B (Acc 27.68 and RE 2.4).

We notice that Galactica and LLaMA split numbers into individual tokens. For
example $123.456$ is converted into $1\ 2\ 3\ .\ 4\ 5\ 6$. Razeghi et al.
(2022) show that arithmetic ability is related to pre-training term
frequencies. For tokens that appear more in pre-training, LLMs can have better
accuracy in answering arithmetic expressions about them. Number tokens with
more digits (e.g. 23) apparently appear less than single digit token (e.g. 2
and 3). Splitting numbers into individual tokens neglects all number tokens
with more digits and makes all single digit tokens (mainly $0\sim 9$) appear
in the pre-training corpus in the same order of magnitude. Galactica-30B and
LLaMA-30B obtain 45.14 and 30.17 in terms of accuracy (list in Table 3) that
outperforms OPT-30B (15.96), Bloom-176B (20.2), and GLM-130B (25.94), which
show superiority of digit-level tokenization.

###  4.3 Training

#### Self-supervised Pre-training

While pre-training, code corpus and LaTeX-sources are possible to relate to
arithmetic ability since they all contain arithmetic operators and numbers.
Code-davinci-002 is pretrained with code corpus. Code-davinci-002 performs
well on many reasoning-related tasks Zhou et al. (2022), however, it performs
not good compared to other LLMs in arithmetics. This proves that mathematical
reasoning ability is different from arithmetic ability which needs to
understand numbers deeply. Galactica with numerous LaTeX-sources outperforms
other LLMs except for InstructGPT and ChatGPT which show LaTeX is useful.

#### Instruction Tuning

is also very important in arithmetic ability. Comparing Opt-30B (Acc 15.96 RE
2.28 NNR 11.22) with Opt-Iml-Max-30B (Acc 17.46 RE 1.52 NNR 6.23), Bloom (Acc
20.2 RE 2.6 NNR 18.45) with BloomZ (Acc 22.44 RE 1.5 NNR 4.74), and code-
davinci-002 (Acc 21.7) with text-davinci-002 (Acc 42.89) in Table 3 show that
instruction tuning can boost the performance in all metrics. Text-davinci-003
(RLHF) outperforms text-davinci-002 (SFT) in arithmetic tasks which shows RLHF
is important for building arithmetic ability.

###  4.4 Prompts

#### Input Prompts

We find the best prompts are different across LLMs. We list the best and worst
prompts for LLMs in Table 8. We find models are sensitive to input prompts and
not using prompts is the worst option for most LLMs. For InstructGPT and
ChatGPT, using “Calculate” as a prompt perform best. For other LLMs, using
LaTeX-related prompts perform best.

Model | Best | Acc | Worst | Acc
---|---|---|---|---
gpt-3.5-turbo-0301 | Cal* | 75.06 | $$ | 64.59
text-davinci-003 | Cal | 56.61 | Eqa | 43.64
galactica-120b | Eqa | 45.14 | $\emptyset$ | 38.9
llama-65b | Eqa | 28.43 | Cal | 4.74
opt-175b | Cal | 21.7 | $\emptyset$ | 15.21
gpt-neox-20b | Eqa | 35.41 | $\emptyset$ | 26.93
glm-130b | $ | 25.94 | $\emptyset$ | 22.44
bloomz-176b | $$ | 22.44 | $\emptyset$ | 11.72
Table 4: Best and worst prompts for different LLMs.

#### System Prompts

For ChatGPT, we can also provide system-level messages as instruction prompts.
Table 5 shows providing system-level messages improves ChatGPT’s accuracy and
reduces relative error significantly. The most different groups are group 13
irrational numbers and group 16 logarithm functions. Without a system-level
message, ChatGPT thinks $e$ can be Euler’s number or a variable and cannot
give an answer. For logarithm functions, ChatGPT tries to explain how it
calculates which may mislead our provided parser. We notice that if we require
ChatGPT to output results to four decimal places, it will have a zero non-
number ratio. To conclude, ChatGPT will try to explain the calculation
procedure without a system-level prompt and will only provide answers with a
system-level prompt.

Group | Cal | Cal* | Cal*4
---|---|---|---
| Acc | RE | Acc | RE | Acc | RE
0 Euler | 100 | .00 | 100 | .00 | 100 | .00
$1\sim 6$ $+-$ | 97 | .00 | 96 | .00 | 93 | .01
$7\sim 10$ $\times\div$ | 69 | .20 | 69 | .01 | 71 | .01
$11\sim 12$ $\wedge$ | 50 | .24 | 50 | .32 | 50 | .27
$13$ Irr. | 64 | 1.73 | 72 | .56 | 84 | .11
$14$ Long | 68 | .19 | 64 | .46 | 60 | .59
$15$ Tri. | 44 | 1.21 | 48 | .96 | 44 | 1.40
$16$ Log | 56 | .80 | 60 | .04 | 56 | .01
Overall | 74 | .33 | 75 | .14 | 74 | .14
Table 5: Comparing different system prompts in ChatGPT on MATH 401. Cal means no system prompt. * = “You are an accurate calculator.” 4 = “Calculating to four decimal places.” Expression |  Calculate: |  Zero-shot COT
---|---|---
(88 - 42) * (33 * 41) - (13 - 27) = 62252 |  First, we need to simplify the expression inside the parentheses: 88 - 42 = 46 33 * 41 = 1353 13 - 27 = -14 Now we can substitute these values into the original expression: (46) * (1353) - (-14) Multiplying 46 and 1353 gives: 62,238 Remember that subtracting a negative number is the same as adding a positive number, so we can rewrite the expression as: 62,238 + 14 Adding 14 gives: 62,252 Therefore, the answer is: 62,252 |  First, we simplify the parentheses: (88 - 42) * (33 * 41) - (13 - 27) = 46 * 1353 - (-14) Next, we simplify the second set of parentheses by changing the subtraction into addition of the opposite: = 46 * 1353 + 14 We can then multiply 46 and 1353: = 62338 \+ 14 Finally, we add: = 62352 Therefore, (88 - 42) * (33 * 41) - (13 - 27) = 62352.
Table 6: Example of Group 14 decoded by ChatGPT with Calculate and COT
prompts.

###  4.5 Interpolation and Extrapolation

LLMs have strong abilities to fit on in-domain data. If pretraining corpora
contain arithmetic expressions, it is easy for LLMs to memorize them. For out-
of-domain data, LLMs need to extrapolate how to calculate them. We do not know
what are in-domain data and out-of-domain data for models (especially
ChatGPT), so it is hard to test their interpolation and extrapolation
abilities. We use the easy group and the hard group to estimate the
interpolation and extrapolation abilities. The easy group queries have
possibilities that appear in the pretraining corpora or instruct-tuning, while
the hard group queries contain big numbers / decimal numbers / long
expressions which are very unlikely to be covered by pretraining corpora or
instructions. Thus answering easy queries may examine the interpolation
ability of models and answering hard queries must examine the extrapolation
ability of the models. We find ChatGPT performs best on hard queries, and all
other models have limited performance on hard queries which show limited
extrapolation.

Figure 1: Performances of MATH 401 on LLMs with different sizes. We do not
know the parameter count of ChatGPT. We list InstructGPT results with SFT
setting (text-davinci-002) only for a fair comparison.

###  4.6 Scaling Laws

To understand how parameter counts influence arithmetic ability, we plot the
results with different-size LLMs in Figure 1. We do not plot text-davinci-003,
gpt-3.5-turbo-0301 and gpt-4 since they do not have smaller versions with the
same setting. We find that LLMs have better abilities with larger parameter
counts. An interesting phenomenon we found is model over 30B does not improve
significantly compared with 30B models, especially in Galactica where the 120B
model performs the same as the 30B model. We hypothesize that 30B may be
enough for arithmetic ability. ChatGPT may be a model smaller than 175B which
outperforms other 175B models a lot, thus larger parameter count does not
guarantee better arithmetic ability. For GPT-4, we cannot have any possible
guess. Considering its much slower inference speed, we guess it has larger
parameter counts than ChatGPT and obtain better reasoning ability (i.e. long
arithmetic expression).

###  4.7 Chain-of-Thought

LLMs can leverage chain-of-thought to better answer math word problems Wei et
al. (2022b). We test on ChatGPT whether chain-of-thought will improve
arithmetic calculations. We use the prompt “Let us solve this equation step by
step” to instruct ChatGPT for zero-shot COT Kojima et al. (2022). We compare
the results of zero-shot COT using “Calculate:” in Table 7. Surprisingly, we
find that COT does not improve the performance of any group even in group 14
with long arithmetic expressions. To understand the reason for this
phenomenon, we check decoded results for these two prompts in Table 6. We find
using “Calculate:” as the prompt can automatically generate chain-of-thoughts
for long arithmetic expressions and generate answers directly for easy
questions.

Group | Cal | 0 COT
---|---|---
| Acc | RE | Acc | RE
0 Euler | 100 | .00 | 100 | .00
$1\sim 6$ $+-$ | 97 | .00 | 94 | .02
$7\sim 10$ $\times\div$ | 69 | .20 | 61 | .66
$11\sim 12$ $\wedge$ | 50 | .24 | 48 | .56
$13$ Irr. | 64 | 1.73 | 28 | 4.89
$14$ Long | 68 | .19 | 64 | .46
$15$ Tri. | 44 | 1.21 | 40 | 1.14
$16$ Log | 56 | .80 | 28 | 5.37
Overall | 74 | .33 | 66 | .98
Table 7: Comparing zero-shot COT and Calculate using ChatGPT on MATH 401. Model | Naive | ICL
---|---|---
| Acc | RE | Acc | RE
galactica-120b | 45.14 | 1.3 | 45.14 | 0.42
galactica-6.7b | 34.41 | 2.61 | 32.67 | 0.65
flan-t5-xxl | 3.74 | 5.78 | 0.0 | 10.0
flan-t5-base | 2.49 | 3.18 | 0.0 | 10.0
Table 8: In-context learning on MATH 401.

###  4.8 In-context Learning

In-context learning (ICL) provides related question-answer pairs to improve
LLMs Brown et al. (2020); Wei et al. (2022b). In our task, we can provide
similar arithmetic expressions before the queries to help model understanding
the arithmetic operator as done in Smith et al. (2022). We provide 8 similar
cases (we promise these cases are different from the query) for each query. We
test whether ICL can improve the well-behaved model (Galactica) and the
underperforming model (Flan-T5). For Galactica, it does not improve accuracy
but reduces relative error significantly. For small-sized Flan (smaller than
3B) it cannot generate any number under the setting of in-context-learning.

##  5 Conclusion

In this paper, we propose MATH 401 to evaluate the arithmetic ability of LLMs.
We find that tokenization, pre-training corpus, prompts, and model parameter
counts are important for their arithmetic ability. The reason ChatGPT performs
so well in arithmetic still has some mystery, i.e. the parameter counts and
instruction datasets of ChatGPT. We hope this paper can help readers improve
LLMs with better arithmetic ability. This paper is only focused on arithmetic,
testing LLMs on other math topics including symbolic mathematics, solving
(ordinary differential, partial differential) equations, calculus, algebra,
geometry, probability theory, and graph theory are also interesting topics.

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##  Appendix A Examples from MATH 401

We list examples for each group from MATH 401.

  * •

$e^{i\pi}+1=0$

  * •

$5+9=14$

  * •

$21+97=118$

  * •

$721-847=-126$

  * •

$714637232158-667119914538=47517317620$

  * •

$-1+(-6)=-7$

  * •

$-0.038+0.0092=-0.0288$

  * •

$78\times 64=4992$

  * •

$5.0\times 0.09=0.045$

  * •

$45960\times 59693=2743490280$

  * •

$70\div 61=1.1475$

  * •

$7^{4}=2401$

  * •

$2.242^{3.7342}=20.3865$

  * •

$e+\pi=5.8598$

  * •

$(4\times 64)\times(39+12)=13056$

  * •

$\sin(-3.75\pi)=0.7071$

  * •

$\log_{10}(797)=2.9015$