# The Strong Gauss Lucas Theorem and Analyticity of Correlation Functions via
the Lee-Yang Theorem

Barry Simon1 Dedicated to the memory of Freeman Dyson.

###### Abstract.

We provide a simple mechanism for going from Lee-Yang type theorems to
analyticity of correlation functions by exploiting under appreciated
inequalities of Newman. We also describe a Lee-Yang approach that recovers the
consequences of a low density cluster expansion for spin $S$ models without
any combinatorics.

###### Key words and phrases:

Ising Model, Correlation Functions, Lee-Yang, Gauss-Lucas, Cluster Expansions

###### 2020 Mathematics Subject Classification:

Primary: 82B20, 20C15, 30C15 ;Secondary: 41A58, 30B40

1 Departments of Mathematics and Physics, Mathematics 253-37, California
Institute of Technology, Pasadena, CA 91125, USA. E-mail<EMAIL_ADDRESS>
##  1\. Introduction

Freeman Dyson was a master of a large swaths of modern theoretical and
mathematical physics with important contributions. He returned several times
to the area I’d call the theory of lattice gases, i.e. the Ising and classical
Heisenberg models. Notable are his famous series [4, 5, 6] on the existence of
phase transitions in slowly decaying $1D$ Ising models which also introduced
the hierarchical models which turn out to be especially useful in mathematical
understanding of the renormalization group.

On a more personal level, there is the joint papers [7, 8] he wrote with
Elliott Lieb and me. One of the high points of my time in Princeton [37] were
the weekly several hour meetings the three of us had in Freeman’s office in
the first few months of 1976 discussing many aspects of spin systems leading
to our papers which contain what remains the only rigorous results on
continuous system breaking in a quantum statistical mechanics model. So it
seemed appropriate to provide this memorial with some remarks on analyticity
in classical lattice gases.

Our main subject here concerns proving analyticity of correlation functions of
Ising models as a function of magnetic field using Lee-Yang methods. This was
first addressed by Lebowitz-Penrose [17] in 1968 who were able to prove it in
the spin $1/2$ case. In 1974, Newman [26] extended the Lee-Yang theorem to an
optimal class of single spin distributions. It appears that it wasn’t until
2012 that Fröhlich-Rodriguez [9] proved analyticity of correlations in this
generality; they had a second paper [10] on cluster expansions and decay of
correlations in this generality. One of our main points here is the remark
that Newman [26] could have proven this result by rather different methods
using an inequality he proved but didn’t use, namely for
$\operatorname{Re}(h)>0$, one has that

$\operatorname{Re}\left(\frac{f_{\Lambda}(j_{1},\dots,j_{n};h)}{f_{\Lambda}(j_{1},\dots,j_{n-1};h)}\right)>0$
(1.1)

where

$f_{\Lambda}(j_{1},\dots,j_{n};h)\equiv\langle\sigma_{j_{1}}\dots\sigma_{j_{n}}\rangle_{\Lambda,h}$
(1.2)

with $\langle\cdot\rangle_{\Lambda,h}$ the free BC Ising expectation in
magnetic field $h$.

The short version of this paper is the remark that while the Vitali
convergence theorem [38, Theorem 6.2.8] is usually stated assuming the
analytic functions, $g_{n}$, are uniformly bounded on compacts, it is valid if
one merely has one-sided bounds on the real parts of $g_{n}$; the simplest way
to see this is to note that if say $\operatorname{Re}(g_{n})\geq 0$, then
$h_{n}=e^{-g_{n}}$ are uniformly bounded, so we can apply Vitali to the
$h_{n}$ and Hurwitz’ Theorem [38, Theorem 6.4.1] to see that their limit is
non-vanishing which implies convergence of the $g_{n}$. From this observation,
it is a few lines to conclude convergence and so analyticity of the
correlation functions.

Rather than stop with this punchline and a really short paper, I plan to first
provide, in Section 2, the tools needed for a somewhat more direct proof of
(1.1) and a version of the above observation with quantitative bounds (that
thereby provides quantitative bounds on correlations). In Section 3, I will
provide the details of the proof of Newman’s result, (1.1) and of convergence
and bounds on correlations. Section 4 will address a related issue. There
exist (see that Section for references) an extensive literature on using
Ruelle’s extension of Asano’s proof of the Lee-Yang theorem to obtain cluster
expansions for Ising models in the high and low temperature regimes but there
does not seem to be anything similar for the cluster expansion in the large
field (aka large fugacity) region even though this doesn’t require the
somewhat involved group theoretic considerations of the work on high and low
temperatures. Since it is reasonable to have these results (which avoid any
combinatorial estimates) in the literature, I sketch them in Section 4.

##  2\. Fun and Games with Gauss Lucas

The Gauss-Lucas theorem (named after Lucas [22, 23, 24], whose earliest result
was in 1868, and Gauss who never published it but had it in his letters and
notebooks as early as 1835) asserts that if $P(z)$ is a polynomial, then the
complex roots of $P^{\prime}$, lie in the convex hull of the complex roots of
$P$. The simplest proof follows from the formula

$f(z)\equiv\frac{P^{\prime}(z)}{P(z)}=\sum_{j=1}^{n}\frac{1}{z-z_{j}}$ (2.1)

if

$P(z)=A\prod_{j=1}^{n}(z-z_{j})$ (2.2)

from which the complex conjugate of $f(w)=0$ implies that (note that if
$P^{\prime}(w)=0$, then either $w$ is equal to some $z_{j}$ or else $f(w)=0$)

$w=\sum_{j=1}^{n}a_{j}z_{j};\qquad
a_{j}\equiv\frac{|w-z_{j}|^{-2}}{\sum_{k=1}^{n}|w-z_{k}|^{-2}}$ (2.3)

The more common proof relies on what we’ll call the strong Gauss-Lucas
Theorem:

######  Theorem 2.1 (Strong Gauss-Lucas Theorem).

If $P$ is a non-constant complex polynomial which is non-vanishing on
${\mathbb{H}}_{+}\equiv\\{z\,\mid\,\operatorname{Re}(z)>0\\}$, then one has
that $f$ given by (2.1) obeys

$z\in{\mathbb{H}}_{+}\,\Rightarrow\,\operatorname{Re}(f(z))>0$ (2.4)

In particular, $P^{\prime}$ is non-vanishing on ${\mathbb{H}}_{+}$.

######  Remark.

Once one has this, one sees that if any given open half plane is free of zeros
of $P$, it is free of zeros of $P^{\prime}(z)$. Since the convex hull of the
zeros of $P$ is the complement of the union of all half planes free of zeros,
this implies the Gauss-Lucas theorem.

###### Proof.

If $\operatorname{Re}(z_{j})\leq 0$ and $z\in{\mathbb{H}}_{+}$, then
$\operatorname{Re}(1/(z-z_{j}))>0$, so (2.1) implies (2.4). ∎

Following Lieb-Sokal [21], we define the space ${\mathcal{A}}_{a}$ for any
$a\geq 0$, as the space of entire functions with $\lVert f\rVert_{b}<\infty$
for all $b>a$ where

$\lVert f\rVert_{b}=\sup_{z}e^{-b|z|^{2}}|f(z)|$ (2.5)

${\mathcal{A}}_{a}$ is a countable normed Fréchet space with the set of norms
$\lVert f\rVert_{a+1/n}$. One reason that it is better to deal with this
Fréchet space rather than the Banach space where is a single norm is finite is
the freedom of being able to wiggle the value of $b$ in
$\lVert\cdot\rVert_{b}$ gives us, as is seen by the following easy to prove
fact

######  Proposition 2.2.

(a) The Taylor series of any $f\in{\mathcal{A}}_{b}$ (some $b\geq 0$) converge
to $f$ in the topology of ${\mathcal{A}}_{b}$.

(b) Let $\\{f_{m}\\}$ be a sequence which is bounded in ${\mathcal{A}}_{b}$
(some $b\geq 0$), i.e., for each $c>b$, we have $\sup_{m}\lVert
f_{m}\rVert_{c}<\infty$. Suppose the $f_{m}$ converges on a set with a limit
point. Then $f_{m}$ has a limit in ${\mathcal{A}}_{b}$.

The spaces ${\mathcal{A}}_{a}$ have analogs for functions of $\nu$ complex
variables. We define $\lVert f\rVert_{b}$ for entire functions,
$f(z_{1},\dots,z_{\nu})$, of $\nu$ variables by

$\lVert f\rVert_{b}=\sup\\{e^{-b\sum_{j=1}^{\nu}|z_{j}|^{2}}|f(z)|\\}$ (2.6)

${\mathcal{A}}_{a}({\mathbb{C}}^{\nu})$ is the space of functions with $\lVert
f\rVert_{b}<\infty$ for all $b>a$. Proposition 2.2 extends easily to these
spaces.

We also define ${\mathcal{P}}^{\nu}$ to be the set of polynomials,
$P(z_{1},\dots,z_{\nu})$, of $\nu$ variables which are non-vanishing if
$Re(z_{j})>0$ for $j=1,\dots,\nu$ (we denote this set of $\mathbf{z}$ by
${\mathbb{H}}_{+}^{\nu}$) and we let ${\mathcal{P}}_{a}^{\nu}$ be its closure
in ${\mathcal{A}}_{a}({\mathbb{C}}^{\nu})\setminus\\{f\equiv 0\\}$. If
$\nu=1$, we will sometimes drop the superscript. Since convergence in
${\mathcal{A}}_{a}$ implies convergence uniformly on compacts, Hurwitz’
theorem implies that if $f\in{\mathcal{P}}_{a}^{\nu}$, then $f$ is non-
vanishing on ${\mathbb{H}}_{+}^{\nu}$. However we note that the converse is
false for (see [21]) the function $z\mapsto e^{bz^{2}};b>0$ is non-vanishing
on ${\mathbb{H}}_{+}$ but does not lie in any ${\mathcal{P}}_{a}$.

One key to the proof of (1.1) will be (we use $\partial_{j}$ as shorthand for
$\frac{\partial}{\partial z_{j}}$)

######  Theorem 2.3.

(a) For each $\nu$, $a>0$ and $j=1,\dots,\nu$, the map $\partial_{j}$ is a
bounded map of ${\mathcal{A}}_{a}({\mathbb{C}}^{\nu})$ to itself.

(b) If $f\in{\mathcal{P}}^{\nu}_{a}$ and $\partial_{j}f$ is not identically
zero, then $\partial_{j}f\in{\mathcal{P}}^{\nu}_{a}$ for $j=1,\dots,\nu$.

(c) If $f\in{\mathcal{P}}^{\nu}_{a}$ with $\partial_{j}f$ not identically
zero, then on ${\mathbb{H}}_{+}^{\nu}$, we have that

$\frac{\partial|f|^{2}}{\partial
x_{j}}(z)=2|f(z)|^{2}\operatorname{Re}\left(\frac{\partial_{j}f(z)}{f(z)}\right)>0$
(2.7)

###### Proof.

(a) By symmetry, we can suppose that $j=1$. By a Cauchy estimate,

$\partial_{1}f(\mathbf{z})\leq\lVert
f\rVert_{b-\varepsilon,\nu}\exp\left((b-\varepsilon)\left[(|z_{1}|+1)^{2}+\sum_{k=2}^{\nu}|z_{k}|^{2}\right]\right)$
(2.8)

We can find $C$ so that for all $y>0$, we have that
$2(b-\varepsilon)y\leq\varepsilon y^{2}+C$, so with
$G=\exp(C+(b-\varepsilon)^{2})$, we have that

$\partial_{1}f(\mathbf{z})\leq G\lVert
f\rVert_{b-\varepsilon,\nu}\exp\left(b\left[\sum_{k=1}^{\nu}|z_{k}|^{2}\right]\right)$
(2.9)

For any $b>a$, pick $\varepsilon=\tfrac{1}{2}(b-a)$ to get
$\lVert\partial_{1}f(\mathbf{z})\rVert_{b}\leq G\lVert
f\rVert_{b-\varepsilon,\nu}$ which proves (a).

(b) By the Gauss-Lucas theorem applied to the polynomial
$P(\cdot,z_{2},\dots,z_{\nu})$, one sees that if $P\in{\mathcal{P}}^{\nu}$,
then so is $\partial_{1}P$ so by (a), if $P_{n}\in{\mathcal{P}}^{\nu}$
converges to $f$ in ${\mathcal{A}}_{a}$, then $\partial_{1}P_{n}$ converges to
$\partial_{1}f$ proving (b).

(c) By (2.4) applied to the polynomial $P(\cdot,z_{2},\dots,z_{\nu})$, one
sees that if $P\in{\mathcal{P}}^{\nu}$, then
$\operatorname{Re}(\partial_{1}P/P)>0$ on ${\mathbb{H}}_{+}^{\nu}$. Taking
limits, one sees the final inequality in (2.7). For the first equality, we
note that

$\displaystyle 2\operatorname{Re}\left(\frac{\partial_{j}f(z)}{f(z)}\right)$
$\displaystyle=\frac{\partial_{j}f(z)}{f(z)}+\frac{\overline{\partial_{j}f(z)}}{\overline{f(z)}}$
$\displaystyle=\frac{\overline{f(z)}\partial_{j}f(z)+f(z)\overline{\partial_{j}f(z)}}{|f(z)|^{2}}$
$\displaystyle=|f(z)|^{-2}(\partial_{j}+\overline{\partial_{j}})(f\overline{f}(z))$
(2.10) $\displaystyle=|f(z)|^{-2}\frac{\partial|f|^{2}}{\partial x_{j}}(z)$

where to get (2), we used
$\overline{\partial_{j}}f=\partial_{j}(\overline{f})=0$ (a form of the Cauchy-
Riemann equations; see [38, Problem 2.1.2]). ∎

Later we will need a form of Theorem 2.3 for spin $1/2$ that goes back to the
disk rather than the half plane.

######  Lemma 2.4.

Let $f(z)=Az+Bz^{-1}$ and suppose that for some $R>0$, we have that $f(z)\neq
0$ if $|z|<R$. Then

$\operatorname{Re}\left(\frac{z\frac{\partial f}{\partial
z}}{f(z)}\right)<0\text{ and so }z\frac{\partial f}{\partial z}(z)\neq 0$
(2.11)

for $|z|<R$.

###### Proof.

The case $A=0$ is trivial. If $A\neq 0$, the ratio in (2.11) only depends on
$B/A$, so without loss, we suppose that $A=1$. In that case, the condition of
not vanishing if $|z|<R$ is equivalent to $|B|\geq R^{2}$. Noting that
$z\tfrac{\partial f}{\partial z}=z-Bz^{-1}$, we compute

$\displaystyle\operatorname{Re}\left(\frac{z\frac{\partial f}{\partial
z}}{f(z)}\right)$
$\displaystyle=\operatorname{Re}\left[\frac{z-Bz^{-1}}{z+Bz^{-1}}\right]$
$\displaystyle=\frac{\operatorname{Re}[(z-Bz^{-1})(\bar{z}+\overline{B}\bar{z}^{-1})}{|z+Bz^{-1}|^{2}}=\frac{|z|^{2}-|B|^{2}|z|^{-2}}{|z+Bz^{-1}|^{2}}<0$
(2.12)

when $|z|<R$ since $|B|\geq R^{2}$. ∎

######  Theorem 2.5.

(a) Let $F(z_{1},\dots,z_{\nu})$ by a function on
$({\mathbb{C}}\setminus\\{0\\})^{\nu}$ of the form

$F(z_{1},\dots,z_{\nu})=\sum_{\sigma_{1}=\pm 1,\dots,\sigma_{\nu}=\pm
1}a(\sigma_{1},\dots,\sigma_{\nu})z_{1}^{\sigma_{1}}\dots
z_{\nu}^{\sigma_{\nu}}$ (2.13)

and suppose that $F(z_{1},\dots,z_{\nu})\neq 0$ if
$|z_{1}|<R_{1},\dots,|z_{\nu}|<R_{\nu}$. Then for any $\ell$ and
$k_{1},\dots,k_{\ell}\in\\{1,\dots,\nu\\}$, we have that
$\prod_{j=1}^{\ell}z_{k_{j}}\frac{\partial}{\partial z_{k_{j}}}f(z)\neq 0$ if
$|z_{1}|<R_{1},\dots,|z_{1}|<R_{\nu}$ and on that set

$\operatorname{Re}\left(\frac{\prod_{j=1}^{\ell}z_{k_{j}}\frac{\partial}{\partial
z_{k_{j}}}f(z)}{\prod_{j=1}^{\ell-1}z_{k_{j}}\frac{\partial}{\partial
z_{k_{j}}}f(z)}\right)<0$ (2.14)

(b) Let $Z$ be the function on ${\mathbb{C}}^{\nu}$ given by

$Z(h_{1},\dots,h_{n})=\sum_{\sigma_{1}=\pm 1,\dots,\sigma_{\nu}=\pm
1}a(\sigma_{1},\dots,\sigma_{\nu})\exp\left(\sum_{j=1}^{\nu}h_{j}\sigma_{j}\right)$
(2.15)

and suppose that $Z(h_{1},\dots,h_{\nu})\neq 0$ if
$\operatorname{Re}(h_{1})>A_{1},\dots,\operatorname{Re}(h_{\nu})>A_{\nu}$.
Then for any $\ell$ and $k_{1},\dots,k_{\ell}\in\\{1,\dots,\nu\\}$, we have
that $\frac{\partial^{\ell}}{\partial h_{k_{1}}\dots\partial
h_{k_{\ell}}}Z(h)\neq 0$ if
$\operatorname{Re}(h_{1})>A_{1},\dots,\operatorname{Re}(h_{\nu})>A_{\nu}$ and
on that set

$\operatorname{Re}\left(\frac{\frac{\partial^{\ell}}{\partial
h_{k_{1}}\dots\partial h_{k_{\ell}}}Z(h)}{\frac{\partial^{\ell-1}}{\partial
h_{k_{1}}\dots\partial h_{k_{\ell-1}}}Z(h)}\right)>0$ (2.16)

######  Remark.

(b) is of course a consequence of the proof of Theorem 3.2 below in case the
apriori measure is the spin $1/2$ Ising measure and one can get (a) from (b)
by the change of variables we use to go in the other direction. So this is an
alternate proof of that special case.

###### Proof.

(a) follows from the Lemma and induction since we can fix all the variables
but the one we are taking the derivative of. If $z_{j}=e^{-h_{j}}$, then
$z_{j}\tfrac{\partial}{\partial z_{j}}=-\tfrac{\partial}{\partial h_{j}}$ so
with $R_{j}=e^{-A_{j}}$, (a) implies (b). ∎

We end this section with the promised quantitative version of the remark on
Vitali under only control of the real part (even though it has no relation to
Gauss-Lucas). One way is to use a basic result from complex analysis, the
Borel-Carathédory theorem [38, Problem 3.6.12], that if $f$ is analytic on the
unit disk ${\mathbb{D}}$ and continuous on its closure and $0<r<1$, then

$\max_{|z|\leq
r}|f(z)|\leq\frac{2r}{1-r}\max_{|z|=1}\operatorname{Re}(f(z))+\frac{1+r}{1-r}|f(0)|$
(2.17)

From this and a simple covering argument, one easily shows if $z_{0}\in
K\subset\Omega$ with $K$ compact and $\Omega$ open, there is a constant $C$
(depending only on $z_{0}$, $K$ and $\Omega$) so that for all $f$ analytic on
$\Omega$, one has that

$\sup_{z\in K}|f(z)|\leq
C\left(|f(z_{0})|+\sup_{z\in\Omega}\operatorname{Re}(f(z))\right)$ (2.18)

Instead, we will use the Herglotz representation for Caratheédory functions
[39, Theorem 5.4.1], i.e. an analytic function, $g$, on ${\mathbb{D}}$ with
$g(0)=1$ and $\operatorname{Re}g\geq 0$, has the form:

$g(z)=\int\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\mu(e^{i\theta})$ (2.19)

for a probability measure, $d\mu$, on $\partial{\mathbb{D}}$. Since
$\max_{\theta}(|1+re^{-i\theta}|)=1+r$ and
$\min_{\theta}(|1-re^{-i\theta}|)=1-r$, applying this to $g=f/f(0)$, one
concludes that

######  Theorem 2.6.

If $f$ is analytic on ${\mathbb{D}}$ with $\operatorname{Re}(f(z))>0$ there
and $\operatorname{Im}(f(0))=0$, one has that

$|f(z)|\leq f(0)\,\frac{1+|z|}{1-|z|}$ (2.20)

We will be interested in functions analytic with positive real part on
${\mathbb{H}}_{+}$, so we conformally map ${\mathbb{H}}_{+}$ to
${\mathbb{D}}$:

######  Theorem 2.7.

If $f$ is analytic on ${\mathbb{H}}_{+}$ with $\operatorname{Re}(f(h))>0$
there and $\operatorname{Im}f(1)=0$, then for all $h\in{\mathbb{H}}_{+}$, we
have that

$\alpha(h)^{-1}f(1)\leq|f(h)|\leq\alpha(h)f(1);\qquad\alpha(h)\equiv\frac{|1+h|+|1-h|}{|1+h|-|1-h|}$
(2.21)

###### Proof.

Define

$z(h)=\frac{1-h}{1+h}$ (2.22)

Then, $z$ maps the imaginary axis to the unit circle, has $z(1)=0$ and is a
bijection of the Riemann sphere to itself so it maps ${\mathbb{H}}_{+}$
biholomorphically to ${\mathbb{D}}$. Let $g$ be defined on ${\mathbb{D}}$ so
that $g(z(h))=f(h)$. Since $|z(h)|=|1-h|/|1+h|$, one sees that

$\frac{1+|z|}{1-|z|}=\alpha(h)$ (2.23)

so the second inequality in (2.21) is just (2.20). By noting that $f(h)^{-1}$
also has a positive real part, we can apply the second inequality to
$f(h)^{-1}$ to get the first inequality. ∎

##  3\. Convergence and Analyticity of Correlations

Here are the models we want to discuss. We start with an even probability
measure, $\mu$, on ${\mathbb{R}}$, called the apriori measure, which obeys

$\int e^{Ax^{2}}\,d\mu(x)<\infty\text{ for all }A>0$ (3.1)

Given a finite set $\Lambda\subset{\mathbb{Z}}^{\nu}$, we let
$\langle\cdot\rangle_{0,\Lambda}$ be the expectation in the product measure
$\otimes_{k\in\Lambda}d\mu(x_{j})$ on ${\mathbb{R}}$. Fix a symmetric matrix
$\\{J_{k\ell}\\}_{k,\ell\in\Lambda}$ with

$J_{k\ell}\geq 0$ (3.2)

and form the Hamiltonian

$H(\\{x_{k}\\}_{k\in\Lambda})=-\sum_{k,\ell\in\Lambda}J_{k\ell}x_{k}x_{\ell}$
(3.3)

For $\mathbf{h}\in{\mathbb{C}}^{\Lambda}$, we are interested in the function
(easily seen to be an entire function on ${\mathbb{C}}^{\Lambda}$):

$Z(\mathbf{h})=\langle\exp(-H(x)-\sum_{k\in\Lambda}h_{k}x_{k})\rangle_{0,\Lambda}$
(3.4)

especially on the set ${\mathbb{H}}_{+}^{\Lambda}$. Important is the Lee-Yang
property of not vanishing on ${\mathbb{H}}_{+}^{\Lambda}$. That this is true
when $d\mu$ is the spin $1/2$ Ising measure,
$\tfrac{1}{2}(\delta_{+1}+\delta_{-1})$, is the celebrated Lee-Yang circle
theorem [20] (the name comes from the fact that they used the variable
$z=e^{2\beta h}$ for which their result shows what is in their case the
polynomial $z^{|\Lambda|/2}Z(h_{j}\equiv h)$ has all its zeros on the unit
circle). The Lee-Yang theorem is important because of the realization of Lee-
Yang [44] that this property, convergence of $(Z_{\Lambda})^{1/|\Lambda|}$
when all $h_{j}=h$ real and the Vitali theorem prove that the pressure (or
free energy per unit volume depending on how the model is interpreted) is real
analytic for $h>0$ and indeed has an analytic continuation to all of
${\mathbb{H}}_{+}$.

Newman [26] found an optimal result specifying those $\mu$ for which the Lee-
Yang property holds. A PN measure is an even probability measure on
${\mathbb{R}}$ obeying the condition that

$E_{\mu}(z)=\int\,e^{zx}\,d\mu(x)$ (3.5)

is non-vanishing whenever $\operatorname{Re}(z)>0$ (and so also when
$\operatorname{Re}(z)<0$ since $E_{\mu}(-z)=E_{\mu}(z)$.). I choose the name
after Newman and Pólya [29, 30] (Pólya got interested in which even measures
had Laplace transforms with only imaginary zeros as part of an unsuccessful
attempt to prove the Riemann hypothesis). In particular, Pólya proved that the
measure $N^{-1}e^{-A\cosh(x)}\,dx$ is a PN measure. Pólya’s approach to the
Riemann hypothesis was extended by deBruijn [3] and Newman [27]. Not all
measures are PN measures; a direct calculation shows that the three point
measure,
$\tfrac{\lambda}{2}(\delta_{+1}+\delta_{-1})+(1-\lambda)\delta_{0};0<\lambda\leq
1$ is a PM measure if and only if $\lambda\geq 1/3$. But all measures of
special interest in statistical mechanics are PN measures: this includes
(equal weight) spin $S$ (either by a simple direct calculation or Griffiths
[11]), the distribution of the first component of a unit vector
equidistributed on a $D$-sphere (whose Fourier transform is well known to be a
Bessel function, all of whose zeros are real), $N^{-1}e^{-A\cosh(x)}\,dx$
(done by Pólya, as noted), and $N^{-1}\exp(-ax^{4}+bx^{2})\,dx$ (by Griffiths-
Simon [12] or as noted by Newman [26] as a scaled limit of Pólya’s example).

A moment’s thought shows that a measure has the Lee-Yang property when all
$J_{k\ell}=0$ if and only if it is a PN measure. Newman [26] made the
remarkable discovery that this necessary condition for the Lee-Yang property
for all ferromagnetic $J$ is also sufficient. Lieb-Sokal [21] found an
alternate proof and more importantly the following stronger result.

######  Theorem 3.1.

If $\mu$ is a PN measure, then for all $J_{k\ell}\geq 0$, the function $Z$ of
(3.4) lies in ${\mathcal{P}}^{|\Lambda|}_{a=0}$ and, in particular, is non-
vanishing on ${\mathbb{H}}_{+}^{|\Lambda|}$.

######  Remark.

[21] prove a stronger result that obtains some results for apriori measures in
some ${\mathbb{R}}^{m}$ but the quoted result has a simpler proof which they
prove in an appendix (as well as from their more general result), and suffices
for what we need here.

We turn next to the correlation functions. We fix a translation invariant pair
interaction $J(j-k)\geq 0$ (with $J(-j)=J(j)$) and for $h>0$ define

$f_{\Lambda}(j_{1},\dots,j_{n};h)\equiv\langle\sigma_{j_{1}}\dots\sigma_{j_{n}}\rangle_{\Lambda,h}$
(3.6)

where $\langle\cdot\rangle_{\Lambda,h}$ is the free BC state with pair
interaction, external magnetic field $h$ and apriori measure $d\mu$ at each
site. We’ll also define

$J=\sum_{j\in{\mathbb{Z}}^{\nu}}J(j)$ (3.7)

which we suppose is finite.

It is a fundamental consequence of the analyticity guaranteed by Theorem 3.1
(Ruelle [33], Lebowitz-Martin-Löf [16]) that when $d\mu$ is a PN measure,
there is a unique equilibrium state which is the limit of the
$\langle\cdot\rangle_{\Lambda,h}$. We use $f(j_{1},\dots,j_{n};h)$ for this
limit. (The uniqueness result requires that $\text{\rm{supp}}(\mu)$ is
compact. In general [19], one only gets a unique tempered state - the limit of
the free BC is tempered. Since it is peripheral, we’ll ignore this issue; the
reader can either supply details or assume the support is compact).

From Theorem 3.1 and the methods of Section 2, we get the main result of this
note:

######  Theorem 3.2.

Let the single site distribution be a PN measure. The infinite volume limits,
$f(j_{1},\dots,j_{n};h)$, have analytic continuations to the region
$\operatorname{Re}(h)>0$ and obey

$\alpha(h)^{-n}L^{n}\leq|f(j_{1},\dots,j_{n};h)|\leq\alpha(h)^{n}Q_{n}$ (3.8)

where $\alpha$ is given by (2.21), $Q_{n}$ is an explicit $d\mu$ dependent
constant,

$L=\frac{\int xe^{x}\,d\mu(x)}{\int e^{x},d\mu(x)}$ (3.9)

Moreover, the finite volume correlations converge to this analytic function
for all $h$ with $\operatorname{Re}(h)>0$.

######  Remarks.

1\. We emphasize that the upper bounds in (3.8) depend only on $n$ and $h$ and
are uniform the $j_{k}$’s. This is useful in proving $m\geq 0$ for the
argument in [18]. In general, we will get the upper bounds at $h=1$ using
Ruelle [35], but, of course if $d\mu$ has compact support with convex hull,
$[-S,S]$, we can use the use the trivial bound $Q_{n}=S^{n}$.

2\. By the uniqueness of state for real $h$ we get that the limits exist for
$\operatorname{Re}(h)>0$ for any BC where the Lee-Yang theorem is applicable
(so the finite volume expectation has non-vanishing denominator), e.g.
periodic BC.

###### Proof.

The derivatives of $Z_{\Lambda}(\\{h_{\ell}\\}_{\ell\in\Lambda})$ are given by

$\frac{\partial^{n-1}}{\partial h_{j_{1}}\dots\partial
h_{j_{n-1}}}Z=Z\langle\sigma_{j_{1}}\dots\sigma_{j_{n-1}}\rangle$ (3.10)

where the expectation is with a $j$ dependent $h_{j}$. So by Theorem 3.1 and
(2.7), we conclude inductively that
$\langle\sigma_{j_{1}}\dots\sigma_{j_{n-1}}\rangle$ is non-vanishing when
$h\in{\mathbb{H}}_{+}$ and in that region, one had that

$\operatorname{Re}\left(\frac{f_{\Lambda}(j_{1},\dots,j_{n};h)}{f_{\Lambda}(j_{1},\dots,j_{n-1};h)}\right)>0$
(3.11)

By Theorem 2.7 and Vitali’s theorem, the proof of our theorem is reduced to
proving that

$L^{n}\leq f_{\Lambda}(j_{1},\dots,j_{n};h=1)\leq Q_{n}$ (3.12)

By GKS inequalities (Kelly-Sherman [15] or Simon [40, Chapter 2]), we have
that
$f_{\Lambda}(j_{1},\dots,j_{n};h=1)\geq\prod_{k=1}^{n}f_{\Lambda}(j_{k};h=1)\geq(\langle\sigma\rangle_{0,h=1})^{n}=L^{n}$
where $\langle\cdot\rangle_{0,h=1}$ is the expectation of a single spin in
external field $h=1$. By Holder’s inequality

$f_{\Lambda}(j_{1},\dots,j_{n};h=1)\leq\prod_{k=1}^{n}(f_{\Lambda}(j_{k},\dots,j_{k};h=1))^{1/n}$

Under our assumptions, the system obeys all the requirement of Ruelle [35] who
proves [35, Theorem 2.2] explicit apriori bounds on probabilities that imply
bounds on $f_{\Lambda}(j_{k},\dots,j_{k};h=1)$, uniformly in $\Lambda$; see
[40, Section 2.3]. ∎

##  4\. A Poor Person’s Large Field Cluster Expansion

The Lee-Yang idea [44] that tracking zeros can be used to prove analyticity
can also be used to provide the results of cluster expansions without any
combinatorial estimates at all. As explicated by collaborations around Gruber
and Slawny (some of their basic papers are [13, 25, 41, 42, 14, 43]) this can
be done for both the high temperature and ferromagnetic low temperature
regions. Remarkably, there does not seem to be in the literature an explicit
version of this for the large field (aka large fugacity, low density or high
density) region even though as we’ll see it is quite simple without the need
for the involved group theoretic analysis of the high and low temperature
exapnsions. The one big limitation compared to the more usual cluster
expansions is that the analysis is restricted to spin $1/2$ (or equal weight
spin $S$).

Here is the framework we’ll use. At each point, $j\in{\mathbb{Z}}^{\nu}$, we
have a $\pm 1$ Ising spin, $\sigma_{j}$. For any finite subset
$A\subset{\mathbb{Z}}^{\nu}$, we define

$\sigma^{A}=\prod_{j\in A}\sigma_{j}$ (4.1)

and, as usual, $\langle\cdot\rangle_{0,\Lambda}$ is the product of equal
weight Bernoulli expectation of spins in a finite set
$\Lambda\subset{\mathbb{Z}}^{\nu}$.

We fix $J_{0}(A)\geq 0$ for all $A$ with $\\#(A)\geq 2$ with two properties:
it is translation invariant and the collection, ${\mathcal{A}}$, of those $A$
with $J_{0}(A)\neq 0$ is finite range in the sense that

$q=\\#\\{A\,\mid\,A\in{\mathcal{A}},A\ni 0\\}<\infty$ (4.2)

We let $v$ be the number of equivalence class under translations of
$A\in{\mathcal{A}}$ (so $2v\leq q$, since if $A\in{\mathcal{A}}$, we have that
$\\#(A)$ translates of it containing $0$).

Below, when we write $J(A)$, we will mean possible complex numbers which are
translation invariant (we will suppose that $J(A)=0$ if
$A\notin{\mathcal{A}}$). In the usual way (Ruelle [31], Simon [36]), for real
parameters, $J(A)$ and $h$, one forms the finite volume Hamiltonian, partition
function and pressure

$\displaystyle-H_{\Lambda}$
$\displaystyle=\sum_{A\subset\Lambda}J(A)\sigma^{A}+h\sum_{j\in\Lambda}\sigma_{j}$
$\displaystyle Z_{\Lambda}=\langle e^{-H}\rangle_{0,\Lambda}$
$\displaystyle\qquad p=\lim|\Lambda|^{-1}\log(Z_{\Lambda})$ (4.3)

and one defines equilibrium states via the DLR equations. Finally, we define

$I_{0}=\max_{A\in{\mathcal{A}}}2^{\\#(A)}e^{2J_{0}(A)}$ (4.4)

Here is what we’ll prove:

######  Theorem 4.1.

Given an interaction as just defined, there is a unique translation
equilibrium invariant state and the pressure and all correlation functions are
jointly analytic on the open set in ${\mathbb{C}}^{v+1}$ given by
$\\{h\,\mid\,|e^{-2h}|<1/qI_{0}\\}\times\\{J(A)\,\mid\,A\in{\mathcal{A}},|J(A)|<J_{0}(A)\\}$.
For this unique translation invariant equilibrium state, the mass gap defined
by

$m=\limsup_{|k-\ell|\to\infty}\left\\{-\frac{1}{|k-\ell|}\log\left[\langle\sigma_{k}\sigma_{\ell}\rangle-\langle\sigma_{k}\rangle\langle\sigma_{\ell}\rangle\right]\right\\}$
(4.5)

is strictly positive.

######  Remarks.

1\. More precisely, there is a unique translation invariant equilibrium state
when all parameters are real and in the larger set, one has joint analyticity
in the set described.

2\. These are the major results one gets from cluster expansions in the large
$h$ region for spin $1/2$. In particular, it has the initial results needed to
prove joint analyticity in $(\beta,h)$ [17] and mass gap [18] in the full
$\operatorname{Re}(h)>0$ for pair interacting ferromagnetic Ising models (see
Simon [40, Sections 3.7-3.8]).

3\. By Griffiths [11], a spin $S$ equal weight model has the same states,
pressure, etc as an analog spin $1/2$ model so by considering that model and
fixing the coupling within the spin $1/2$ spins which sum to a spin $S$, one
can extend this theorem to the spin $S$ situation.

We will use Ruelle’s [32, 34] extension of the Asano contraction theorem [2]

######  Proposition 4.2 (Ruelle-Asano Theorem).

Let $\Lambda$ be a finite set, let
$z_{\Lambda}\equiv\\{z_{x}\\}_{x\in\Lambda}$ be the coordinates of a point in
${\mathbb{C}}^{|\Lambda|}$ and for $X\subset\Lambda$, define

$z^{X}=\prod_{x\in X}z_{x}$ (4.6)

Let $\\{\Lambda_{\alpha}\\}_{\alpha\in A}$ be a finite cover of $\Lambda$ and
for each $\alpha\in A$, a polynomial,

$P_{\alpha}(z_{\Lambda_{\alpha}})=\sum_{X\subset\Lambda_{\alpha}}c_{X}^{(\alpha)}z^{X}$
(4.7)

Suppose for each $\alpha\in A$ and $x\in\Lambda$, we have a closed set
$M_{x}^{(\alpha)}\subset{\mathbb{C}}\setminus\\{0\\}$ so that if $z_{x}\notin
M_{x}^{(\alpha)}$ for all $x\in\Lambda_{\alpha}$, we have that
$P_{\alpha}(z_{\Lambda_{\alpha}})\neq 0$. Let

$P(z_{\Lambda})=\sum_{X\subset\Lambda}\left(\prod_{\alpha\,\mid\,\Lambda_{\alpha}\cap
X\neq\emptyset}c_{\Lambda_{\alpha}\cap X}^{(\alpha)}\right)z^{X}$ (4.8)

Then $P(z_{\Lambda})\neq 0$ if for all $x\in\Lambda$, one has that

$z_{x}\notin-\prod_{\alpha\,\mid\,x\in\Lambda_{\alpha}}(-M_{x}^{(\alpha)})$
(4.9)

Rather than make use of the formula for the final coefficients in (4.8), we’ll
use the Asano contraction intuition that leads to it. Multiaffine polynomials
like (4.7) arise in the Lee-Yang [44] scheme by replacing
$h\sum_{j\in\Lambda}\sigma_{j}$ by $\sum_{j\in\Lambda}h_{j}\sigma_{j}$,
multiplying $Z_{\Lambda}$ by $\exp(\sum_{j\in\Lambda}h_{j}\sigma_{j})$ and
writing the result as a function of $z_{j}=e^{2h_{j}}$. Asano contraction
results from forcing two spins, say, $\sigma_{k}$ and $\sigma_{\ell}$ to be
parallel, i.e. dropping the terms with $\sigma_{k}=-\sigma_{\ell}$ and
replacing $h_{k}\sigma_{k}+h_{\ell}\sigma_{\ell}$ by a single $h\sigma$ term.
Given the $P_{\alpha}$’s, one introduces variables
$\\{z_{\alpha,x}\\}_{x\in\Lambda_{\alpha}}$, forms
$\prod_{\alpha}P_{\alpha}(\\{z_{\alpha,x}\\}_{x\in\Lambda_{\alpha}})$ then
contracts for each $x\in\Lambda$ pairwise all the $z_{\alpha,x}$ with
$x\in\Lambda_{\alpha}$. What results is (4.8) and the Proposition just tracks
its zeros.

We’ll need two lemmas to get the mass gap.

The first is an elementary piece of complex analysis:

######  Lemma 4.3.

Let $f$ be analytic in ${\mathbb{D}}_{R}(0)$ with
$C=\sup_{z\in{\mathbb{D}}_{R}(0)}|f(z)|<\infty$. Suppose that $f^{(k)}(0)=0$
for $k=0,\dots,K-1$. Then for any $z\in{\mathbb{D}}_{R}(0)$, we have that

$|f(z)|\leq\frac{C(|z|/R)^{K}}{1-(|z|/R)}$ (4.10)

###### Proof.

Let $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be the Taylor expansion about $z=0$
for $f$. By a Cauchy estimate, [38, Theorem 3.1.8]

$|a_{n}|\leq CR^{-n}$ (4.11)

By the hypothesis on derivatives, the sum in the Taylor series starts at
$n=K$. We can sum the bounds in (4.11) using a geometric series to get (4.10).
∎

The Ursell functions are defined by

$u_{n}(X_{1},X_{2},\dots,X_{n})=\left.\frac{\partial^{n}}{\partial
h_{1}\cdots\partial
h_{n}}\log\left\langle\exp\left(\sum_{j=1}^{n}h_{j}X_{j}\right)\right\rangle\right|_{h_{j}=0}$
(4.12)

######  Lemma 4.4.

Let $\langle\cdot\rangle$ be a product measure on single site distributions on
$\Lambda\subset{\mathbb{Z}}^{\nu}$ (could be an infinite set). For any finite
set, $A\subset\Lambda$, let $\operatorname{diam}(A)=\max_{j\neq m\in A}|j-m|$.
Let $A_{1},\dots,A_{k}$ be $k$ sets each with $\operatorname{diam}(A_{j})\leq
R$. Let $p,\ell\in\Lambda$ so that $|p-\ell|>kR$. Then the Ursell function
$u_{k+2}(\sigma_{p},\sigma_{\ell},\sigma^{A_{1}},\dots,\sigma^{A_{k}})=0$.

###### Proof.

If $|p-\ell|>kR$, an easy geometric argument proves that one can break
$\\{1,\dots,k\\}$ into two sets $P$ and $Q$ so that $\\{p\\}\cup_{j\in
P}A_{j}$ is disjoint from $\\{\ell\\}\cup_{j\in Q}A_{j}$. It is a basic fact
(sometimes called the second Percus axiom (Percus [28]) that if a set of
variables can be broken into two independent pieces, its Ursell function
vanishes; this follows immediately from (4.12) if we note that the $\log$ is a
sum of $\log$s because of independence. We conclude that the specified
$u_{k+2}$ is zero. ∎

###### Proof of Theorem 4.1.

For each set finite $\Lambda\subset{\mathbb{Z}}^{\nu}$, we define (with
$z^{B}=\prod_{j\in B}z_{j}$)

$Z_{\Lambda,J(A)}(z)=\sum_{A\subset\Lambda}z^{A}p(A);\qquad
p(A)=\prod_{\mathclap{\begin{subarray}{c}B\subset A\\\ \\#(B)>1,\text{
odd}\end{subarray}}}e^{-2J(B)}$ (4.13)

which with $z_{j}=e^{-2h_{j}}$ is the (analytic continuation of the) partition
function in $j$ dependent field. $A$ is the set of negative spins and $p(A)$
the Gibbs factor for $A$.

We model our proof on the Asano proof of the Lee-Yang circle theorem. We
define, for each $A$ with $A\in{\mathcal{A}}$, as a polynomial of $|A|$
variables $\\{z_{j,A}\\}_{j\in A}$

$R_{A}(\\{z_{j,A}\\}_{j\in A})=\sum_{\mathclap{\begin{subarray}{c}B\subset
A\\\ \text{ even}\end{subarray}}}\left(\prod_{j\in
B}z_{j,A}\right)+e^{-2J(A)}\sum_{\mathclap{\begin{subarray}{c}B\subset A\\\
\text{ odd}\end{subarray}}}\left(\prod_{j\in B}z_{j,A}\right)$ (4.14)

The term for $B=\emptyset$ is $1$, and, if $r\equiv\max_{j}|z_{j,A}|\leq 1$,
all other terms are bounded by $e^{J_{0}(A)}r$, so
$|R_{A}(z)-1|<2^{|A|}e^{J_{0}(A)}r\leq rI_{0}$, so if $r<1/I_{0}$, we see that
$R_{A}(z)\neq 0$.

We can get $Z_{\Lambda}(\\{z_{j}\\}_{j\in\Lambda})$ by taking
$\prod_{A\subset\Lambda}R_{A}(\\{z_{j,A}\\}_{j\in A})$ (i.e. copies of $z_{j}$
for each $A\ni j$) and Asano contracting all the copies of $z_{j,A}$ together
to a single $z_{j}$. Since there are at most $q$ copies of each $z_{j}$, we
see that, by Proposition 4.2, $Z$ is non-vanishing if all $z_{j}$ obey
$|z_{j}|<1/qI_{0}\equiv R_{0}$. It will be important later (not to control
pressure but to control correlation functions), that the same is true of the
$Z_{per}$ defined with periodic BC, i.e. if we take $\Lambda$ to be a
hypercube and take all $A$ with $J(A)\neq 0$ with $A\subset\Lambda$ and take
all translates in $\Lambda$ when we connect it to a torus.

By the usual Lee-Yang argument [44] (Vitali’s convergence theorem), this
proves joint analyticity of the pressure in the limit (for $z_{j}$ all taken
equal). Fix some real value of all $J(A)$ finite range. Now pick a $B$ with
$B\notin{\mathcal{A}}$ and form $Z_{B,J_{B}}(z)$. When $J(B)=0$,
$Z_{B}(z)=\prod_{j\in B}(1+z_{j})$ is non-vanishing when all
$z_{j}\in{\mathbb{D}}$. By continuity of the zeros, it follows that for any
$\rho<1$, we can find $\varepsilon(\rho)$ so that if
$|J(B)|<\varepsilon(\rho)$, then $Z_{B,J_{B}}(z)\neq 0$ so long as
$z\in{\mathbb{D}}_{\rho}(0)$. If we now form $Z_{\Lambda}$ with an additional
of $J(B)$ term, then $Z\neq 0$ so long as
$|z_{j}|<\rho^{p}R_{0}=\rho^{p}/qI_{0}$ where $p=\\#(B)$, for we get this from
the $J(B)=0$ situation by an additional Asano contraction of all
$Z_{C,J_{B}}(z)$ with $C$ a translation of $B$ that lies inside $\Lambda$ and
each $z_{j}$ is involved in at most $p$ such contractions. We thus see,
because $\rho$ can be taken arbitrarily close to $1$, that there is a complex
neighborhood (in ${\mathbb{C}}^{|\Lambda|+1}$) in the set of complex $z_{j}$
with $|z_{j}|<1/I_{0}q$ and $J(B)=0$ where the new $Z$ is non-vanishing. It
follows that for any real $h$ with $e^{-2h}<1/qI_{0}$, that the pressure is
real analytic near $J(B)=0$, so, by [36, Theorem III.3.11],
$\langle\sigma^{B}\rangle$ is the same in all translation invariant
equilibrium states. We also have analyticity in $J(A)$ where $J_{0}(A)\neq 0$.
Since $B$ was arbitrary, we conclude there is a unique such state!

Because any limit point of the periodic BC states is a translation invariant
equilibrium state, we conclude that every such limit point is the unique
translation invariant equilibrium state and thus we have convergence to that
state. By Theorem 2.5, we see that in the region of joint analyticity in $h$
and $J(A)$ described above, for any $j_{1},\dots,j_{\ell}$ in a torus,
$\Lambda$, we have that

$\operatorname{Re}\left(\frac{\langle\sigma_{j_{1}}\dots\sigma_{j_{\ell}}\rangle_{\Lambda}}{\langle\sigma_{j_{1}}\dots\sigma_{j_{\ell-1}}\rangle_{\Lambda}}\right)>0$
(4.15)

where the expectation is with periodic BC. Fix real values of the parameters.
Since any limit point of the periodic BC state is a translation invariant
equilibrium state and such a state is unique, we have convergence for such
real parameters. By Theorem 2.7 and (4.15) by induction, we get uniform (in
$\Lambda$) bounds on the correlation functions in the region of analyticity,
so by Vitali’s theorem we get convergence and analyticity of the infinite
volume limit.

Fix some real $h$ in the region of analyticity. Put a $\beta$ in front of all
of the other couplings. If we show $m>0$ for all small real positive $\beta$,
then by the method of Lebowitz-Penrose [18], it is strictly positive
everywhere in the region of analyticity. Because of Lemma 4.3 and the uniform
bounds on
$\langle\sigma_{k}\sigma_{\ell}\rangle-\langle\sigma_{k}\rangle\langle\sigma_{\ell}\rangle$
that follow from (4.15) and (2.21), it suffices to prove that if, for some
$R>0$, $|k-\ell|>QR$ implies that the $j$th derivative of
$\langle\sigma_{k}\sigma_{\ell}\rangle-\langle\sigma_{k}\rangle\langle\sigma_{\ell}\rangle$
with respect to $\beta$ at $\beta=0$ vanishes for $j<Q$. Since the limit of
states at $\beta=0$ is uncoupled (and so independent) single sites, this
follows with $R$ the range of the interaction because of Lemma 4.4 and (4.12)
which shows that the $j$th derivative in question is a sum of Ursell functions
of the form
$u_{q+2}(\sigma_{k},\sigma_{\ell},\sigma^{A_{1}},\dots,\sigma^{A_{k}})$ with
each $A_{j}\in{\mathcal{A}}$. ∎

######  Remark.

Instead of getting analyticity of correlations from (4.15), one could use the
proof of uniqueness of state - it implies joint analyticity of the pressure
and so analyticity of the derivative which is the correlation.

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# A Unique Approach to Classify Inflationary Potentials

Somnath<EMAIL_ADDRESS>and P K<EMAIL_ADDRESS>
_School of Physics, University of Hyderabad_
_Central University P.O, Hyderabad - 500046. India._

###### Abstract

Inflationary cosmology has made significant strides in understanding the
physics driving the rapid expansion of the early universe. However, many
inflation models with diverse potential shapes present analysis, comparison,
and classification challenges. In this paper, we propose a novel approach to
tackle this issue. We introduce a general potential formula encompassing all
inflationary potentials, whether single-field or multi-field, into a single
mathematical framework. This formula establishes a unified framework for
systematically classifying inflation models based on their potential
functions. We showcase the efficacy of the general potential formula by
successfully reproducing well-known inflation models, such as the Starobinsky
potential and the Valley Hybrid Inflation model. Moreover, we derive general
inflationary parameters, including the slow-roll parameters and power spectra,
using the proposed formula. Our approach provides a versatile tool for
classifying and studying various inflationary scenarios, simplifying the
analysis and comparison of different models in the field of inflationary
cosmology.

##  1 Introduction

The study of inflationary cosmology has revolutionized our understanding of
the early universe, providing a compelling framework to explain the observed
isotropy, homogeneity, and flatness [1]. Inflationary models posit a period of
rapid expansion in the early universe, driven by a homogenous scalar field
known as inflaton [2]. These models have successfully addressed the
shortcomings of the standard Big Bang cosmology, such as the horizon and
flatness problems, and have provided predictions that align with a wide range
of cosmological observations [2]. While inflationary cosmology has made
remarkable progress, the field remains rich with many inflation models, each
characterized by different potentials for inflation. The diversity of these
potentials arises from the underlying physics and the dynamics of the early
universe, giving rise to many inflationary scenarios. This variety poses
challenges in analyzing, comparing, and classifying different models,
hindering our ability to gain deeper insights into the fundamental physics
driving inflation. Inflationary cosmology has witnessed numerous efforts to
classify and categorize inflation models based on their underlying physics and
potential shapes. These classification schemes aim to capture the diversity of
inflationary potentials and provide a systematic framework for understanding
the range of inflation scenarios proposed in the literature.

One common approach to classifying inflationary models is based on the shape
of the inflation potential. This method categorizes inflation potentials into
power-law potentials, exponential potentials, hybrid potentials, [2] etc. Each
class represents a specific functional form for the potential, often motivated
by specific underlying physics or symmetry considerations. While this approach
provides a straightforward categorization, it can be limited in its ability to
capture the full diversity of inflation models, as it may overlook subtle
variations within each class. Another classification method involves
characterizing inflationary models based on their observational predictions.
This approach considers inflation predictions for various cosmological
observables, such as the spectral index of primordial fluctuations, the
tensor-to-scalar ratio, and the non-Gaussianity parameter [2]. Models that
yield similar predictions for these observables are grouped, forming a
classification based on the resulting observational signatures [3]. This
method is valuable for connecting inflation models to empirical data such as
the CMB observations. Still, it may need to fully capture the potential shapes
that give rise to these predictions.

In addition to the classification schemes based on the shape of the inflation
potential and observational predictions, another important categorization in
the field of inflationary cosmology is based on the energy scales involved in
the inflationary process. This classification distinguishes high-field and
low-field inflation models [3]. Along this line, various physical properties
might be considered to classify the inflationary potentials. In this paper, we
propose a novel approach to address this challenge by presenting a general
potential formula that encompasses all inflationary potentials. By allowing
the functional form of the potential to be chosen, our approach offers
flexibility in capturing the diversity of inflationary scenarios. It unifies
all the inflationary potentials, single-field or multi-field, under one
functional form.

##  2 General Potential

The following comprehensive general formula encapsulates the dynamics of
inflationary models with multiple fields. Our formula offers a unified
approach, enabling the seamless incorporation of any inflationary model into a
single mathematical framework.

$V(\varphi_{m})=M^{4}\bar{A}(\varphi_{m})\left[\bar{B}(\varphi_{m})+\sum_{i=0}^{n}a_{i}\bar{C_{i}}(\varphi_{m})\right]^{p}.$
(1)

Here, $a_{0}=0$. $M$ represents a characteristic energy scale associated with
the inflationary dynamics in the inflation potential. It sets the overall
magnitude of the potential energy during inflation. Equation (1) offers a
unified framework for systematically classifying inflation models. The
potential, denoted as $V(\varphi_{m})$, is expressed in terms of a set of
functions $\bar{A}(\varphi_{m})$, $\bar{B}(\varphi_{m})$, and
$\bar{C_{i}}(\varphi_{m})$, where $i$ ranges from $0$ to $n$ and $p$ is any
real number. Here, $m=0,1,2..t$ where $t$ is any integer. $m=0$ corresponds to
the inflaton field $\phi$; other $m$ values can represent other fields. If $m$
is single-valued, it represents single-field inflation, and multi-valued $m$
represents multifield models. We may classify inflationary models based on the
value of $n$, i.e., up to the number $i$ runs in the summation. We can imagine
a series of potentials with various possible functions for each value. In
equation (1), particular $n$ values correspond to the number of terms linearly
associated with the potential, $n=0$ corresponds to single-termed potentials,
$n=1$ corresponds to double-termed potentials, and so on. For the single field
inflation model, $m=0$, and the potential is a function of $\varphi_{o}=\phi$
alone.

$V(\phi)=M^{4}A(\phi)\left[B(\phi)+\sum_{i=0}^{n}a_{i}C_{i}(\phi)\right]^{p}.$
(2)

To demonstrate the effectiveness of the general potential formula for a single
field (2), we will show how a specific choice of the functions and parameters
within the formula reproduces the well-known Starobinsky inflation model. By
selecting $n=1$, $A(\phi)=B(\phi)=1$, $a_{1}=-1$,
$C_{1}(\phi)=e^{-\sqrt{\frac{2}{3}}\frac{\phi}{M_{\text{Pl}}}}$, and $p=2$, we
can obtain the Starobinsky potential [3],[4].

$V(\phi)=M^{4}\left[1-e^{-\sqrt{\frac{2}{3}}\frac{\phi}{M_{\text{Pl}}}}\right]^{2}.$
(3)

Further, in the equation (1) if we choose $n=4$ with
$\bar{A}(\varphi_{m})=1,\bar{B}(\varphi_{m})=\frac{\lambda_{\sigma}}{4},\bar{C_{1}}(\varphi_{m})=\sigma^{4},\bar{C_{2}}(\varphi_{m})=\sigma^{2},\bar{C_{3}}(\varphi_{m})=\phi^{2},\bar{C_{4}}(\varphi_{m})=\phi^{2}\sigma^{2}$
with coefficients
$a_{1}=\frac{\lambda_{\sigma}}{4M^{4}},a_{2}=-\frac{\lambda_{\sigma}}{2M^{2}},a_{3}=\frac{m_{\phi}^{2}}{2M^{4}},a_{4}=\frac{\lambda}{2M^{4}}$,
$m=0,1$ with $\varphi_{0}=\phi$ and $\varphi_{1}=\sigma$, it reproduces the
Valley Hybrid Inflation model [5], [6].

$V(\phi,\sigma)=\frac{\lambda_{\sigma}}{4}(\sigma^{2}-M^{2})^{2}+\frac{1}{2}m_{\phi}^{2}\phi^{2}+\frac{1}{2}\lambda\phi^{2}\sigma^{2}.$
(4)

Also, in equation (1), if we consider $n=1$,
$A(\phi)=-\frac{3}{8l^{2}}\phi^{2}$, $B(\phi)=1$, $a_{1}=-\frac{4\kappa
l^{2}}{9}$, $C_{1}=\phi^{4}$, and $p=1$, it is easy to see that one attains
the effective potential that comes from the AdS Swampland conjectures [7].

$V(\phi)=-\frac{3}{8l^{2}}\phi^{2}+\frac{\kappa}{6}\phi^{6}.$ (5)

It’s apparent that the general potential formulation is highly versatile and
accommodates a wide range of inflationary potentials. It provides a flexible
framework to describe various inflationary models and allows for exploring
different inflationary dynamics.

##  3 Inflationary Parameters

This section explores the implications of the proposed general potential
formula for deriving general inflationary parameters that characterize
inflationary models, as outlined in equation (2). By analyzing the functional
form of the potential and its corresponding dynamics, we can extract essential
parameters that quantify the behavior and observational predictions of
inflation. We drop the scalar field $\phi$ from the expression to ensure
better readability.

The general form of the first slow-roll parameter is

$\displaystyle\epsilon$
$\displaystyle=\frac{p^{2}}{16\pi}\bigg{[}A^{\frac{1}{p}}B+\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}}\bigg{]}^{-2}\bigg{[}\frac{1}{p}A^{\frac{1}{p}-1}A^{{}^{\prime}}B+A^{\frac{1}{p}}B^{{}^{\prime}}+\sum_{i=0}^{n}a_{i}C_{i}^{{}^{\prime}}A^{\frac{1}{p}}+\frac{1}{p}\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}-1}A^{{}^{\prime}}\bigg{]}^{2}.$
(6)

Here, $X^{{}^{\prime}}=\frac{dX}{d\phi}$. The tensor-to-scalar ratio can be
expressed using the first slow roll parameter using the relation,
$r=16\epsilon$. This provides a dynamic approach for choosing the functions by
reflecting the latest bound on the tensor-to-scalar ratio, a crucial
observational quantity in inflationary cosmology. Recent measurements and
cosmic microwave background radiation data analysis have constrained the
tensor-to-scalar ratio. The current bound is $r<0.035$ [8], which might get as
low as $r<0.004$ with future observations [9] [10]. Incorporating these bounds
on $r$ is a valuable guideline while constructing inflationary models using
the proposed general potential formula, allowing us to explore the range of
viable scenarios within the context of current observational constraints. The
general form of the second slow-roll parameter is,

$\displaystyle\eta=\frac{p}{8\pi}\bigg{[}$ $\displaystyle
A^{\frac{1}{p}}B+\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}}\bigg{]}^{-2}\times\bigg{[}(p-1)\Big{(}\frac{1}{p}A^{\frac{1}{p}-1}A^{{}^{\prime}}B+A^{\frac{1}{p}}B^{{}^{\prime}}+\sum_{i=0}^{n}a_{i}C_{i}^{{}^{\prime}}A^{\frac{1}{p}}+\frac{1}{p}\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}-1}A^{{}^{\prime}}\Big{)}^{2}$
(7)
$\displaystyle+\bigg{(}A^{\frac{1}{p}}B+\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}}\bigg{)}\bigg{[}\frac{1}{p}\big{(}\frac{1}{p}-1\big{)}A^{\frac{1}{p}-2}(A^{{}^{\prime}})^{2}B+\frac{1}{p}A^{\frac{1}{p}-1}A^{{}^{\prime\prime}}B+\frac{2}{p}A^{\frac{1}{p}-1}A^{{}^{\prime}}B^{{}^{\prime}}+A^{\frac{1}{p}}B^{{}^{\prime\prime}}$
$\displaystyle+\sum_{i=0}^{n}a_{i}C_{i}^{{}^{\prime\prime}}A^{\frac{1}{p}}+\frac{2}{p}\sum_{i=0}^{n}a_{i}C_{i}^{{}^{\prime}}A^{\frac{1}{p}-1}A^{{}^{\prime}}+\frac{1}{p}\big{(}\frac{1}{p}-1\big{)}\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}-2}(A^{{}^{\prime}})^{2}+\frac{1}{p}\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}-1}A^{{}^{\prime\prime}}\bigg{]}\bigg{]}.$

Let’s consider the Starobinsky potential (3) for testing the general slow roll
parameters. Considering the particular functions mentioned in section 2
applicable for generating Starobinsky potential, we use equation (6) to
produce the first slow roll parameter $\epsilon_{s}$.

$\epsilon_{s}=\frac{1}{6\pi
M_{pl}^{2}}e^{-\sqrt{\frac{2}{3}}\frac{2\phi}{M_{\text{pl}}}}\bigg{[}1-e^{-\sqrt{\frac{2}{3}}\frac{\phi}{M_{\text{pl}}}}\bigg{]}^{-2}.$
(8)

Using the general formula for the second slow roll parameter (7), one can find
the second slow roll parameter for the Starobinsky potential $\eta_{s}$.

$\eta_{s}=\frac{1}{6\pi
M_{pl}^{2}}e^{-\sqrt{\frac{2}{3}}\frac{\phi}{M_{\text{Pl}}}}\bigg{[}4e^{-\sqrt{\frac{2}{3}}\frac{\phi}{M_{\text{Pl}}}}-1\bigg{]}.$
(9)

In this way, the general slow roll parameters can be used to find the slow
roll parameters of any model. The potential (2) can also generalize other
inflationary parameters, such as the scalar power spectrum.

$\displaystyle P_{S}=$
$\displaystyle\frac{128M^{4}}{3p^{2}}\bigg{[}A^{\frac{1}{p}}B+\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}}\bigg{]}^{p-2}\bigg{[}\frac{1}{p}A^{\frac{1}{p}}A^{{}^{\prime}}B+A^{\frac{1}{p}}B^{{}^{\prime}}+\sum_{i=0}^{n}a_{i}C_{i}^{{}^{\prime}}A^{\frac{1}{p}}+\frac{1}{p}\sum_{i=0}^{n}a_{i}C_{i}A^{\frac{1}{p}-1}A^{{}^{\prime}}\bigg{]}^{2}.$
(10)

The tensor power spectrum can be generally expressed as

$\displaystyle P_{T}=$
$\displaystyle\frac{128A}{3}\bigg{[}B+\sum_{i=0}^{n}a_{i}C_{i}\bigg{]}^{p}.$
(11)

The ratio of $P_{T}$ and $P_{S}$ gives us the tensor-to-scalar ratio $r$. We
have derived the general forms of the inflationary parameters considering
$m=0$, which essentially considers the single-field inflationary models. In
the inflationary parameters, if we consider the functions of inflaton field
$A(\phi),B(\phi)$ and $C_{i}(\phi)$ as multi-field functions such as
$\bar{A}(\varphi_{m}),\bar{B}(\varphi_{m})$ and $\bar{C_{i}}(\varphi_{m})$
then these parameters will be valid for multi-field potentials as well. In
such a scenario, the derivatives of the functions will be partial depending on
the model in context.

Various other inflationary parameters can be obtained from the general
potential, such as the other slow-roll parameters, the scalar spectral index,
the running of the indices, and so on.

##  4 Conclusion

The general potential and associated parameters offer a flexible and
comprehensive framework for analyzing various inflationary models. By
employing the general potential, it becomes possible to classify existing
inflationary models within a unified framework. Moreover, this framework
provides a systematic approach to constructing new inflationary models based
on functional considerations. One of the strengths of this general potential
lies in its ability to generate an infinite number of inflationary potentials.
The general form allows for a broad range of functional forms for $A(\phi)$,
$B(\phi)$, and $C_{i}(\phi)$, enabling a vast exploration of potential shapes
and behaviors. This flexibility is crucial for accommodating diverse
inflationary dynamics from different underlying physics. It should be noted
that the general potential is focused on inflation with scalar fields.

However, given the infinite possibilities, it becomes necessary to constrain
the parameter space by selecting a subset of inflationary potentials that
align with current observational data. The aim is to identify those potentials
that successfully reproduce the observed cosmic microwave background
radiation, primordial density fluctuations, and other relevant cosmological
observables. Selecting a subset of inflationary potentials consistent with
data narrows the range of viable models. Through this systematic approach, the
general potential provides a powerful tool for refining existing inflationary
models and discovering new, relevant inflationary scenarios. Focusing on a
functional basis allows us to explore unexplored regions of the parameter
space and discover novel inflationary models that may exhibit unique features
or address current theoretical or observational challenges.

## Acknowledgement

S. D. acknowledges the financial support the Govt of India through the Prime
Minister’s Research Fellowship (PMRF).

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