# Comparison of Ising Models Under Change of Apriori Measure

José Madrid1, Barry Simon2,3, Daniel R. Wells  Dedicated with pleasure to
Elliott Lieb on his $90^{th}$ birthday.

###### Abstract.

We study comparison of correlation functions for ferromagnetic generalized
Ising models with two different apriori measures. One purpose of this note is
to publicize some unpublished 45 year old work of Daniel Wells on the issue.
We then prove results for the apriori measures associated to one component of
$D$-vectors uniformly distributed on the unit sphere and also the case of spin
$S$ ($2S+1$ equally spaced values symmetric about $0$ and with equal weights)
that improves some 50 year old bounds of Griffiths on transition temperatures.

###### Key words and phrases:

Ising Model, Correlation Functions, $D$-vector Model, Spin $S$ Models, Mean
Field Theory, Majorization

###### 2020 Mathematics Subject Classification:

82B20, 26D20, 46N10

1 Department of Mathematics, University of California Los Angeles, Portola
Plaza 520, Los Angeles, CA 90095, USA. E-mail<EMAIL_ADDRESS>
2 Departments of Mathematics and Physics, Mathematics 253-37, California
Institute of Technology, Pasadena, CA 91125. E-mail<EMAIL_ADDRESS>
3 Research supported in part by Israeli BSF Grant No. 2020027.

##  1\. Introduction

Besides Elliott Lieb’s many major accomplishments, there are numerous gems
that sparkle even though they aren’t among his most important. In this note,
we want to discuss something related to his beautiful note [16] on the
infinite spin limit of the pressure of quantum Heisenberg models which
converges to a classical Heisenberg model, a work which motivated the second
author’s extension [20] to more general Lie groups than $SU(2)$. Lieb proved
comparison inequalities for partition functions that squeezed the spin $S$
quantum partition function between the corresponding classical partition
functions with slightly different coupling constants so that the difference of
the coupling constants goes to zero as $S\to\infty$. In the totally
anisotropic case (where only $z$ components are coupled), Dyson, Lieb and the
second author (we never published this work done in 1976 but it was included
it in the 1993 book of Simon [21, Section II.9]) proved monotonicity
(increasing) of the partition function in $S$ and decreasing monotonicity if
the coupling is scaled properly.

When we were working on Thomas-Fermi, Elliott taught the second author about
the magic of convergence of convex functions - that convergence of convex
functions implies convergence of derivatives at points where the limit is
differentiable. This implies convergence of certain correlation functions in
the context that Lieb studied in [16]. But one loses for correlation functions
inequalities like those that Dyson, Lieb and Simon found for partition
functions. It is that question that we want to discuss here. Our framework
will be less general in that we will only consider one component spins and
more general in that we will allow general (even) apriori measures.

The second author is writing a book for Cambridge Press entitled Phase
Transitions in the Theory of Lattice Gases [24]. It is in many ways the
successor to the 1993 book [21] from Princeton University Press. That earlier
book was mainly framework and largely left out all the most fun and beautiful
elements of the theory: Correlation Inequalities, Lee-Yang, Peierls’ Argument,
Berezinskii-Kosterlitz-Thouless transitions and Infrared Bounds which are the
subjects of the new book. But since a different publisher is used, this is
certainly not volume 2 of the earlier work.

The framework for much of the subject is to fix a finite set
$\Lambda\subset{\mathbb{Z}}^{\nu}$, and an apriori even probability measure,
$d\mu$, on ${\mathbb{R}}$, certainly with all moments finite and typically of
compact support.

One considers the configurations in $\Lambda$, i.e. points $\mathbf{\sigma}$
in ${\mathbb{R}}^{\Lambda}$, indicated by $\\{\sigma_{j}\\}_{j\in\Lambda}$ and
uncoupled measure with expectation

$\langle f\rangle_{\mu,0}=\int
f(\mathbf{\sigma})\,\prod_{j\in\Lambda}d\mu(\sigma_{j})$ (1.1)

and one fixes a ferromagnetic Hamiltonian (i.e. $J(A)\geq 0$)

$-H=\sum_{A\subset\Lambda}J(A)\sigma^{A}\qquad\sigma^{A}=\prod_{j\in
A}\sigma_{j}$ (1.2)

or more general over mutliindices, i.e. assignments of an integer, $n_{j}\geq
0$ with then $\sigma^{A}=\prod_{j\in A}\sigma_{j}^{n_{j}}$ (and a finite sum
or else $\ell^{1}$ condition). One then considers, the Gibbs state

$\langle f\rangle_{\mu,\Lambda}=Z^{-1}\langle fe^{-H}\rangle_{\mu,0};\qquad
Z=\langle e^{-H}\rangle_{\mu,0}$ (1.3)

One studies the infinite volume limit with translation invariant $J(A)$,
typically by proving stuff about the finite volume expectations. The
traditional case is the Ising model (aka spin $1/2$ Ising model) where $d\mu$
is a measure supported on $\pm 1$ each point with weight $1/2$; more
generally, we’ll refer to $b_{T}$ with weights $1/2$ at $\pm T$ ($b$ is for
Bernoulli). While a lot of the literature is specific to the spin $1/2$ Ising
model, there is considerable, mathematically interesting, literature on more
general (even) apriori measures. Traditionally, one mainly considered the spin
$S$ measure (for $S=\tfrac{1}{2},1,\tfrac{3}{2},....$, the measure with $2S+1$
pure points equally spaced symmetrically about $0$ and with equal weights) but
the work of Guerra, Rosen and Simon [14] and Griffiths-Simon [13] on discrete
approximations to Euclidean Quantum Field Theory changed that.

As the second author began to write about correlation inequalities in his new
book, he wondered about a natural question. We say that an apriori measure,
$\nu$, on ${\mathbb{R}}$ Ising dominates another measure $\mu$ if and only if
for all $J(A)\geq 0$ and all $B$, one has that

$\langle\sigma^{B}\rangle_{\mu,\Lambda}\leq\langle\sigma^{B}\rangle_{\nu,\Lambda}$
(1.4)

In particular, for general $\mu$ compact support, does one have that $\mu$
Ising dominates $b_{T_{-}}$ and is Ising dominated by $b_{T_{+}}$ for suitable
$0<T_{-}<T_{+}<\infty$. That would imply phase transitions occur for one
apriori measure if and only if they do for all and inequalities on transition
temperatures.

For most, even minor, aspects of the subject of correlation inequalities there
are several papers, sometimes even dozens. So it is surprised that the second
author was unable to find a single published paper on the subject of what we
just called Ising domination! Of course, it was unclear how to search for the
subject in Google. Eventually, we did find one 1978 paper of van Beijeren and
Sylvester [3] that we’ll mention later (see Remark 2 after Theorem 3.1)
although in one respect it is unsatisfactory. And we did also find an appendix
of a paper on another subject but that gets ahead of our story (see the Remark
1 after Theorem 3.1).

One of the pleasant things about writing a book on a subject that one once
knew more about is that one gets to rediscover things that they have
forgotten. With the question of Ising domination in the back of his mind, the
second author found an interesting footnote in a 1980 paper of Aizenman and
er, B. Simon [2]. The footnote said

> then by results of Wells (D. Wells, Some moment inequalities for general
> spin Ising ferromagnets, Indiana Univ. preprint) $\langle
> s_{j}s_{k}\rangle_{\beta,1}\leq
> 2\langle\sigma_{j}^{(1)}\sigma_{k}^{(1)}\rangle_{\beta,2}$.

The left hand side is an Ising expectation and the right with the apriori
measure of the $2D$ rotor with only couplings of the $1$ components. So this
was part of what seems to be an Ising domination result (the subscript $2$
indicates the Ising measure should really be $b_{1/\sqrt{2}}$).

So the second author set about finding this preprint. Google didn’t help
directly but did point him to a 1984 paper of Chuck Newman that mentioned
Wells’ Indiana University PhD. thesis. He wrote to Michael asking if he knew
anything about our footnote and cced Chuck (who had been a grad student with
the second author at Princeton) because the second author conjectured Wells
had been his student. Chuck replied and said he remembered that Wells had been
Slim Sherman’s student. Sherman, the S of GKS and GHS was a delightful
character, long dead.

So the second author wrote to Kevin Pilgrim, the chair at Indiana, who located
a copy of Wells thesis [26] on Proquest. So far though, no one has had any
luck on the preprint nor on locating Wells through Indiana University alumni
records (but see later)! While the thesis did not have anything directly about
the above inequality, it did have a general framework on what we called the
Ising domination problem, lovely material that should have been published.
After an initial draft of this note was written, the first two authors got
some help and located Dr. Wells who kindly agreed to be a coauthor which makes
sense since much of this Note publishes for the first time results from his
thesis. Also, we convinced him to allow us to continue to use the term Wells
domination.

Our main goal in this Note is to describe Wells’ framework in Section 2 and
what we regard as his most significant theorem in Section 3. Since Wells
extended a framework of Ginibre, we begin Section 2 by reminding (telling) you
of that. Then the notion we call Wells’ domination followed by his big theorem
in Section 3 and the notion of canonical lower bound. We’ll note there that
the approach of van Beijeren and Sylvester [3] has one big flaw in that there
is no analog of the Wells Comparison Theorem, Theorem 3.1. Section 4 will then
make explicit the bounds on transition temperatures implied by Wells
Comparison Theorem, recall a result of Griffiths [12] on comparison of
transition temperatures for different spins and note that the bounds when a
measure is canonical are equalities in mean field theory and so optimal in the
high dimension limit. Section 5 has one of our two new results here - that the
distribution of one component of an $D$-vector spin has a canonical lower
bound. From the footnote in [2], it is clear that the missing Wells preprint
has the case $D=2$. We’ll see that case is much easier to prove than $D\geq
3$. In Section 6, we will prove our most significant new result that the spin
$S$ measures for $S\neq 1$ have a canonical lower bound and its consequence
for improving Griffiths bound on transition temperatures for arbitrary spins.
Finally, an appendix A provides the proof of a technical inequality.

We would especially like to thank Kevin Pilgrim for his help in locating
Wells’ thesis [26] and Leonard Schulman and Joshua David Paik for helping the
first two authors make contact with the third. We should also like to thank
Terry Tao. The inequality (6.5) was originally only conjectured by the second
author because it would imply the result on spin $S$ ($S\neq 1$) being
canonical. The second author asked Terry if he could prove the conjecture and
Terry passed it on to the first author (who proved it) thereby making a
successful shidduch.

##  2\. Wells Framework

As mentioned, Wells’ approach is a slight modification of Ginibre’s approach
to the proof of GKS inequalities for Ising type models. In a remarkable 1970
paper [11], Jean Ginibre (who alas passed away in March of 2020 at age 82) not
only found a really simple proof of GKS inequalities but showed somewhat
surprisingly that they held for all (even) apriori measures. If you are new to
Ising models and have time for only one result, this one might be what you
should know.

A Ginibre system is a triple $\langle X,\mu,{\mathcal{F}}\rangle$ of a compact
Hausdorff space, $X$, a probability measure, $\mu$, on $X$ (with expectations
$\langle\cdot\rangle_{\mu}$) and a class of continuous real valued functions
${\mathcal{F}}\subset C(X)$ that obeys:

$(G1)\qquad\forall_{f_{1},...f_{n}\in{\mathcal{F}}}\int_{X}f_{1}(x)\dots
f_{n}(x)\,d\mu(x)\geq 0$ (2.1)
$(G2)\qquad\forall_{f_{1},...f_{n}\in{\mathcal{F}}}\int_{X\times
X}\prod_{j=1}^{n}\left(f_{j}(x)\pm f_{j}(y)\right)\,d\mu(x)d\mu(y)\geq 0$
(2.2)

for all $2^{n}$ choices of the plus and minus sign.

When it is clear which measure is intended, we will drop the $\mu$ from
$\langle\cdot\rangle_{\mu}$. We have restricted to compact Hausdorff spaces
and so bounded functions for simplicity. But since all the arguments are
essentially algebraic, all results extend to the case where $X$ is only
locally compact so long as all $f\in{\mathcal{F}}$ obey
$\int|f(x)|^{m}\,d\mu(x)<\infty$ for all $m$ since that condition assures that
all integrals below are convergent.

Note that

$(G2)\Rightarrow 2\langle
f\rangle_{\mu}=\int_{X}(f(x)+f(y))\,d\mu(x)d\mu(y)\geq 0$

and

$\displaystyle\int_{X\times X}(f(x)-$ $\displaystyle
f(y))(g(x)-g(y))\,d\mu(x)d\mu(y)$ $\displaystyle=2\left[\langle
fg\rangle_{\mu}-\langle f\rangle_{\mu}\langle g\rangle_{\mu}\right]\geq 0$

We will see shortly that $(G2)\Rightarrow(G1)$. What makes the notion so
powerful is that there are three theorems for getting new Ginibre systems from
old ones.

Given a family of functions, ${\mathcal{F}}\subset C(X)$, we define the
Ginibre cone, ${\mathcal{C}}({\mathcal{F}})$, as the set of linear
combinations with non-negative coefficients of products of functions from
${\mathcal{F}}$.

######  Theorem 2.1 (Ginibre Theorem 1).

If a triple $\langle X,\mu,{\mathcal{F}}\rangle$ obeys $(G2)$, so does
$\langle X,\mu,{\mathcal{C}}({\mathcal{F}})\rangle$.

It is trivial that $(G2)$ holds for sums and positive multiples of functions
for which it holds, so it suffices to prove it holds for products. By
induction, we need only handle products of two functions. We note that

$fg\pm f^{\prime}g^{\prime}=\tfrac{1}{2}(f+f^{\prime})(g\pm
g^{\prime})+\tfrac{1}{2}(f-f^{\prime})(g\mp g^{\prime})$ (2.3)

which allows us to prove $(G2)$ for a single product when we have it for
individual functions (and shows (G2)$\Rightarrow$(G1)).

The following is trivial

######  Theorem 2.2 (Ginibre Theorem 2).

Let $\left\\{\langle
X_{j},\mu_{j},{\mathcal{F}}_{j}\rangle\right\\}_{j=1}^{n}$ be a family of
Ginibre systems. Then
$\langle\times_{j=1}^{n}X_{j},\otimes_{j=1}^{n}\mu_{j},\cup_{j=1}^{n}{\mathcal{F}}_{j}\rangle$
is also a Ginibre system

And to add interactions, we use

######  Theorem 2.3 (Ginibre Theorem 3).

Let $\langle X,\mu,{\mathcal{F}}\rangle$ be Ginibre system. Let
$-H\in{\mathcal{F}}$ and define a new measure, $\mu_{H}$ by

$\langle f\rangle_{\mu_{H}}=\frac{\langle fe^{-H}\rangle_{\mu}}{\langle
e^{-H}\rangle_{\mu}}$ (2.4)

Then $\langle X,\mu_{H},{\mathcal{F}}\rangle$ is a Ginibre system.

The proof is easy. The normalization is irrelevant and we expand the
exponential $\exp(-H(x)-H(y))$. Finally

######  Theorem 2.4 (Ginibre Theorem 4).

Let $X$ be ${\mathbb{R}}$ or a compact subset of the form $[-A,A]$ and let
$d\mu$ be a probability measure which is invariant under $x\mapsto-x$ and so
that (only non-trivial in case $X$ is not compact) $\int
x^{2n}\,d\mu(x)<\infty$ for all $n$. Let ${\mathcal{F}}$ contain the single
function, $f(x)=x$. Then $\langle X,\mu,{\mathcal{F}}\rangle$ is a Ginibre
system.

The proof is easy! $(G2)$ says that for all non-negative integers, $k$ and
$m$, one has that

$\int_{X\times X}(x+y)^{k}(x-y)^{m}\,d\mu(x)d\mu(y)\geq 0$ (2.5)

Interchanging $x$ and $y$ implies the integral is zero if $m$ is odd and
$(x,y)\mapsto(-x,-y)$ symmetry implies the integral is zero if $m+k$ is odd.
Thus the only possible non-zero integrals are when $m$ and $k$ are even in
which case the integrand is positive!

A little thought shows that for Hamiltonians of the form

$-H=\sum_{A\subset\Lambda}J(A)\sigma^{A}\qquad\sigma^{A}=\prod_{j\in
A}\sigma_{j}$ (2.6)

with any (!!!) even apriori measure, one has positive expectations and
positive correlations of the $\sigma^{A}$ which is GKS inequalities for
general even measures.

We’d be remiss if we left the subject Ginibre’s wonderful paper without
mentioning two other examples he gives of Ginibre systems that are not
relevant to Wells, although one will appear later. The first is to note that
he proves that if $d\mu$ is a product of rotation invariant measures on
circles, the set of functions $\cos(\sum_{j=1}^{n}m_{j}\theta_{j})$ is a
Ginibre system. This and some extensions are essentially half the correlation
inequalities for plane rotors.

The second is related to an 1882 paper of Chebyshev [6] (which I don’t think
Ginibre knew about when he wrote his 1970 paper) which contained what is
probably the earliest correlation inequality: Chebyshev proved that if $f,g$
are two monotone functions on $[0,1]$, then

$\int_{0}^{1}f(x)g(x)\,dx\geq\int_{0}^{1}f(x)\,dx\int_{0}^{1}g(x)\,dx$ (2.7)

Ginibre proved that for any (not necessarily even) positive probability
measure on ${\mathbb{R}}$, the set ${\mathcal{F}}$ of all positive monotone
functions is a Ginibre family. The proof is again very easy. This is a sort of
poor man’s FKG inequalities.

This completes our review of Ginibre, so we turn to Wells’ work. There is a
simple extension of Ginibre’s method in Wells’ thesis [26] that allows
comparison of measures. Given two probability measures, $\mu$ and $\nu$ on a
locally compact space, $X$, we say that $\mu$ Wells dominates $\nu$, written
$\mu\triangleright\nu$ or $\nu\triangleleft\mu$ with respect to a class of
continuous functions ${\mathcal{F}}$ (with all moments of all
$f\in{\mathcal{F}}$ finite with respect to both measures; not needed if $X$ is
compact) if for all $n$ and all $f_{1},f_{2},\dots,f_{n}$ and all $2^{n}$
choices of $\pm$, we have that

$\int\int(f_{1}(x)\pm f_{1}(y))\dots(f_{n}(x)\pm f_{n}(y))d\mu(x)d\nu(y)\geq
0$ (2.8)

We will be most interested in case $X={\mathbb{R}}$, $\mu$ and $\nu$ are both
even measures with all moments finite and ${\mathcal{F}}$ has the single
function $f(x)=x$ in which case the condition takes the form

$\int_{\mathbb{R}}\int_{\mathbb{R}}(x+y)^{n}(x-y)^{m}d\mu(x)d\nu(y)\geq 0$
(2.9)

for all non-negative integers, $n$ and $m$ in which case we use the symbol
$\triangleleft$ without being explicit about ${\mathcal{F}}$. Since the
measures are even, one need only check this when $n+m$ is even. It is trivial
if both are even, so we only need worry about the case that both are odd.
Since the measures are different, we don’t have the exchange symmetry that
makes the integral vanish if both are odd but symmetry under $y\mapsto-y$
implies invariance under interchange of $m$ and $n$, so we need only check for
$m\geq n$. We’ll see examples later.

Extending the Ginibre machine is effortless. It is easy to prove that

######  Theorem 2.5 (Wells [26]).

(a) If $\mu\triangleleft\nu$ for a set of functions ${\mathcal{F}}$, the same
is true for the Ginibre cone ${\mathcal{C}}({\mathcal{F}})$.

(b) If for $j=1,\dots,n$, $\mu_{j}\triangleleft\nu_{j}$ for probability
measures on spaces $X_{j}$ with respect to sets of functions
${\mathcal{F}}_{j}$ on $X_{j}$, then for the measures on
$\prod_{j=1}^{n}X_{j}$ and the set of functions
$\cup_{j=1}^{n}{\mathcal{F}}_{j}$, one has that
$\otimes_{j=1}^{n}\mu_{j}\triangleleft\otimes_{j=1}^{n}\nu_{j}$.

(c) If $\mu\triangleleft\nu$ for probability measures on a space $X$ with
respect to a set of functions ${\mathcal{F}}$ on $X$, if $-H\in{\mathcal{F}}$
and if $\mu_{H}$, $\nu_{H}$ are Gibbs measures, then
$\mu_{H}\triangleleft\nu_{H}$ for ${\mathcal{F}}$.

(d) If $\mu\triangleleft\nu$ with respect to a set of functions
${\mathcal{F}}$, then for every $f\in{\mathcal{F}}$, we have that

$\int f(x)\,d\mu(x)\leq\int f(x)\,d\nu(x)$ (2.10)

This immediately implies that

######  Corollary 2.6 (Wells [26]).

If for $j=1,\dots,n$, $\mu_{j}\triangleleft\nu_{j}$ for probability measures
on spaces $X_{j}$ with respect to sets of functions ${\mathcal{F}}_{j}$ on
$X_{j}$,then if $-H\in{\mathcal{C}}(\cup_{j=1}^{n}{\mathcal{F}}_{j})$ and if
$\mu_{H},\nu_{H}$ are formed from the underlying product measures
$\otimes_{j=1}^{n}\mu_{j}$ and $\otimes_{j=1}^{n}\nu_{j}$, then for all
$F\in{\mathcal{C}}(\cup_{j=1}^{n}{\mathcal{F}}_{j})$, one has that $\int
f(x)\,d\mu_{H}(x)\leq\int f(x)\,d\nu_{H}(x)$. In particular, if each
$X_{j}={\mathbb{R}}$, (so implicitly $F_{j}$ is the single function
$\sigma_{j}$) and if $H$ has the general ferromagnetic Ising form, (2.6) with
all $J(A)\geq 0$, then for all $A\subset 2^{\\{1,\dots,n\\}}$ one has that

$\langle\sigma^{A}\rangle_{\mu_{H}}\leq\langle\sigma^{A}\rangle_{\nu_{H}}$
(2.11)

Thus by the definition, (1.4), of Ising domination, we see that if
$\mu\triangleleft\nu$, then $\nu$ Ising dominates $\mu$.

Of course, $\triangleleft$ is a binary relation and it is tempting to think of
it as a partial order on measures on ${\mathbb{R}}$ with all moments finite.
Indeed, it is certainly reflexive. It is almost antisymmetric. It is easy to
see that $\mu\triangleleft\nu$ and $\nu\triangleleft\mu$ if and only if $\mu$
and $\nu$ have the same moments. Thus it is antisymmetric among the measures
of compact support or among measures obeying $\int e^{Ax^{2}}\,d\mu(x)<\infty$
for some $A>0$ but not among all measures with finite moments because of the
possibilities of measures non-unique for the moment problem. But we do not
know the following

Question 1 Is Wells relation transitive among all even measures on
${\mathbb{R}}$? How about among all measures on a general topological space if
${\mathcal{F}}$ is rich enough?

Since Ising domination is trivially transitive, for applications, this lack
isn’t so important.

##  3\. The Wells Comparison Theorem

Given an even measure probability, $\mu$ on ${\mathbb{R}}$ and $s>0$, we
define its scaling by

$\mu^{(s)}[A]=\mu[s^{-1}A]$ (3.1)

Then the Bernoulli measure $b_{S}$ defined after (1.3) obeys
$b_{S}=(b_{1})^{(S)}$.

Even if it is not true that $\nu\triangleleft\mu$, it can happen that
$\nu^{(s)}\triangleleft\mu$ for $s$ sufficiently small. In the next section,
we will see that this implies a bound on transition temperatures, so such
comparison results are interesting. The main result of this section implies
that any two non-trivial measures of compact support are comparable in this
sense. Well’s most important result is

######  Theorem 3.1 (Wells Comparison Theorem [26]).

Let $d\mu$ be an even probability measure on ${\mathbb{R}}$ with compact
support that is not a point mass at $0$. Then there are two strictly positive
numbers, $T_{-}(\mu)$ and $T_{+}(\mu)$, so that $\mu\triangleleft b_{S}$ if
and only if $S\geq T_{+}$ and $b_{S}\triangleleft\mu$ if and only if $S\leq
T_{-}$. Moreover

$T_{+}=\sup\\{s\,\mid\,s\in\text{\rm{supp}}(\mu)\\}$ (3.2)

and

$S\leq
T_{-}\iff\forall_{n\in{\mathbb{N}}}\int_{\mathbb{R}}(x^{2}-S^{2})^{n}\,d\mu(x)\geq
0$ (3.3)

######  Remarks.

1\. Bricmont-Lebowitz-Pfister [5] state the existence of $T_{-}$ part of this
theorem, quoting Wells and providing his proof.

2\. There is a very different order from Wells order defined by van Beijeren-
Sylvester [3] (discussed further in [24, Section 2.2]) that also implies Ising
domination but it has the serious flaw that if $0\in\text{\rm{supp}}(\nu)$,
then for no $T>0$ does $\nu$ dominate $b_{T}$ in their order.

3\. The proof below is essentially that of Wells.

######  Lemma 3.2.

Let $\mu$ be a positive measure on an interval $I\subset{\mathbb{R}}$ (either
open or closed at each endpoint). Let $f,g\in L^{2}(d\mu)$ and suppose that
$g$ is monotone increasing on $I$ and there is $c\in I$ so that $f(x)\leq 0$
(resp $f(x)\geq 0$) if $x\leq c$ (resp $x\geq c$). Then

$\int f(x)g(x)\,d\mu(x)\geq g(c)\int f(x)\,d\mu(x)$ (3.4)

###### Proof.

The function $f(x)[g(x)-g(c)]$ is positive so its integral is positive which
is the claim. ∎

###### Proof of Theorem 3.1.

We first prove the existence of $T_{+}$ and (3.2). If
$S\geq\sup\\{s\,\mid\,s\in\text{\rm{supp}}(\mu)\\}$, then, for the integrand
in (2.9) to be positive, we need that
$(S+y)^{n}(S-y)^{m}+(S+y)^{m}(S-y)^{n}\geq 0$ for all $y\geq 0$ in
$\text{\rm{supp}}(\mu)$. If $\mu(\\{0\\})>0$, there is an additional term of
$S^{n+m}\mu(\\{0\\})$ in the right hand side, but that is also positive, so
for such $S$, we have that $\mu\triangleleft b_{S}$.

On the other hand, if $\mu\triangleleft b_{S}$, we have that $\int
x^{2N}\,d\mu(x)\leq S^{2N}$, so, taking $2N$th roots and then $N\to\infty$, we
see that $S\geq\sup\\{s\,\mid\,s\in\text{\rm{supp}}(\mu)\\}$ which proves the
formula for $T_{+}$.

Next we will prove that

$b_{S}\triangleleft\mu\iff\forall_{n\text{
odd}}\int_{\mathbb{R}}(x^{2}-S^{2})^{n}\,d\mu(x)\geq 0$ (3.5)

Taking $n=m$ in the basic integral, we see that

$b_{S}\triangleleft\mu\Rightarrow\forall_{n\text{
odd}}\int_{\mathbb{R}}(x^{2}-S^{2})^{n}\,d\mu(x)\geq 0$ (3.6)

Now look at the basic integral when $\nu=b_{S}$ and $m>n$ with both odd. Since
$(x\pm S)^{n}(x\mp S)^{m}=(x^{2}-S^{2})^{n}(x\mp S)^{m-n}$ we see that the
integral in question is

$\displaystyle\tfrac{1}{2}\int(x^{2}$
$\displaystyle-S^{2})^{n}\left[(x+S)^{m-n}+(x-S)^{m-n}\right]\,d\mu(x)$
$\displaystyle=\int(x^{2}-S^{2})^{n}\left[(x+S)^{m-n}+(x-S)^{m-n}\right]\,d\tilde{\mu}(x)$
(3.7)

where $\tilde{\mu}$ is the measure restricted to $(0,\infty)$ plus
$\tfrac{1}{2}\mu(\\{0\\})\delta_{0}$. By the binomial theorem, the polynomial
$Q_{2k}(y)=(y+S)^{2k}+(y-S)^{2k}$ only has even degree terms with only
positive coefficients so the function in $[\cdot]$ in the last equation is
monotone on $I=[0,\infty)$. Applying the lemma with $c=S$, we see that

$\int_{\mathbb{R}}\int_{\mathbb{R}}(x+y)^{n}(x-y)^{m}d\mu(x)d\nu(y)\geq(2S)^{m-n}\int_{\mathbb{R}}(x^{2}-S^{2})^{n}\,d\mu(x)$
(3.8)

Thus, we have proven (3.5).

Finally, we show that $T_{-}>0$. First, pick $a>0$ so that
$\mu([a,\infty))>0$. Pick $0<b<a$ so small that

$\frac{b^{2}}{a^{2}-b^{2}}\leq\min\left(1,2\mu([a,\infty))\right)$ (3.9)

possible since the left side goes to zero as $b\downarrow 0$. Since the
integrand is positive on $[b,a]$, we have that for all $k\in{\mathbb{N}}$

$\int(x^{2}-b^{2})^{2k+1}d\mu(x)\geq-(b^{2})^{2k+1}+2(a^{2}-b^{2})^{2k+1}\mu([a,\infty))$
(3.10)
$\hbox{}\qquad=2(a^{2}-b^{2})^{2k+1}\left[2\mu([a,\infty))-\left(\frac{b^{2}}{a^{2}-b^{2}}\right)^{2k+1}\right]\geq
0$ (3.11)

by the choice of $b$. Thus $T_{-}\geq b>0$. ∎

One consequence of the theorem is

$T_{-}\leq\left(\int_{\mathbb{R}}x^{2}\,d\mu(x)\right)^{1/2}$ (3.12)

It is an interesting question when one has equality. One would like as good a
lower bound on $T_{-}$ as possible which can yield good lower bounds on
transition temperatures. Often one has equality in (3.12) in which case we
will say that $T_{-}$ is canonical for $\mu$.

######  Example 3.3.

We consider spins taking three values. For $0\leq\lambda\leq 1$, consider the
probability measure supported by the three points $\\{0,\pm 1\\}$ given by

$d\mu_{\lambda}=\tfrac{\lambda}{2}\left(\delta_{1}+\delta_{-1}\right)+(1-\lambda)\delta_{0}$
(3.13)

For $\lambda=2/3$, which is equal weights, this is called (normalized) spin
$1$. Then

$\displaystyle\langle(x^{2}-T^{2})^{2m+1}\rangle_{\lambda}$
$\displaystyle=(1-T^{2})^{2m+1}\lambda-(1-\lambda)T^{2(2m+1)}$
$\displaystyle\geq
0\iff\left[\frac{1-T^{2}}{T^{2}}\right]^{2m+1}\geq\frac{1-\lambda}{\lambda}$
$\displaystyle\iff\frac{1-T^{2}}{T^{2}}\geq\left(\frac{1-\lambda}{\lambda}\right)^{1/2m+1}$
(3.14)

If $\lambda\leq 1/2$, then $(1-\lambda)/\lambda\geq 1$ and the maximum on the
right side of the last formula occurs for $m=0$ while, if $\lambda\geq 1/2$,
then $(1-\lambda)/\lambda\leq 1$ and we get the maximum as $m\to\infty$. Thus,
we find that

$T_{-}(\lambda)=\left\\{\begin{array}[]{ll}\sqrt{\lambda},&\hbox{ if
}\lambda\leq\tfrac{1}{2}\\\ \sqrt{\tfrac{1}{2}},&\hbox{ if
}\lambda\geq\tfrac{1}{2}\end{array}\right.$ (3.15)

So we see there are cases where $T_{-}=\langle
x^{2}\rangle^{1/2}=\sqrt{\lambda}$ and other cases where the inequality is
strict. Note also that at $\lambda=1/2$, the integral
$\langle(x^{2}-T_{-}^{2})^{2m+1}\rangle_{\lambda}$ vanishes for all $n$, a
sign that the distribution of $x^{2}-T_{-}^{2}$ is symmetric about $0$.

In the remainder of this Note, we will discuss bounds on transition
temperatures and then two interesting classes: in Section 5, the distribution
of a single component of a $D$-vector model and in Section 6, the spin $S$
spin.

##  4\. Bounds on Transition Temperatures

Fix a translation invariant ferromagnetic interaction, $J(i-j)\geq 0$ and an
even apriori measure, $\mu$. Let $T_{c}(\mu)$ be the transition temperature
for the model defined as the unique temperature (which may be zero if there is
no phase transition!) so that for larger temperatures, the two point infinite
volume free boundary condition state $\langle\cdot\rangle$ has

$\lim_{j\to\infty}\langle\sigma_{j}\sigma_{0}\rangle=0$ (4.1)

We want to see what $\nu^{(s)}\triangleleft\mu$ implies about the relation of
$T_{c}(\mu)$ and $T_{c}(\nu)$ (a similar analysis holds with other possible
definitions of transition temperature).

The arguments below while stated for Wells order only depend on Ising
domination. Making the temperature and measure explicit, with
$\langle\cdot\rangle_{T,\mu}$ the infinite volume free BC state, we note that
by the definition of $\nu^{(s)}$, we have that (because we are assuming only
pair interactions and because temperature appears as $\sigma_{i}\sigma_{j}/T$)

$\langle\sigma^{A}\rangle_{T,\nu^{(s)}}=s^{|A|}\langle\sigma^{A}\rangle_{T/s^{2},\nu}$
(4.2)

Thus, since Wells order implies Ising domination, we see that
$\nu^{(s)}\triangleleft\mu$ implies that

$\langle\sigma^{A}\rangle_{T,\mu}\geq
s^{|A|}\langle\sigma^{A}\rangle_{T/s^{2},\nu}$ (4.3)

Therefore, if $T\leq s^{2}T_{c}(\nu)$, we see that (4.1) fails for
$\langle\sigma_{j}\sigma_{0}\rangle_{T/s^{2},\nu}$ and so by (4.3) for
$\langle\sigma_{j}\sigma_{0}\rangle_{T,\mu}$. We have thus proven that

######  Proposition 4.1.

Let $\nu^{(s)}\triangleleft\mu$ for two non-trivial even measures. Then

$T_{c}(\mu)\geq s^{2}T_{c}(\nu)$ (4.4)

In particular

$T_{c}(\mu)\geq T_{-}(\mu)^{2}T_{c}(\text{{classical Ising}})$ (4.5)

so if $T_{-}$ is canonical for $\mu$, then

$T_{c}(\mu)\geq\langle x^{2}\rangle_{\mu}T_{c}(\text{{classical Ising}})$
(4.6)

This last putative inequality is especially interesting because the mean field
transition temperature (see, for example, [21, Section II.13] or [24, Section
2.6]) is given by

$T_{MF}(\mu)=\langle x^{2}\rangle_{\mu}\sum_{j}J(j)$ (4.7)

for a pair interacting ferromagnetic model. Thus one has equality in (4.6) if
$T_{c}$ is replaced by $T_{MF}$. It is known [4, 24] that mean field theory is
exact in the infinite dimensional limit of nearest neighbor generalized Ising
models (in the sense that $T_{c}/T_{MF}\to 1$ as $d\to\infty$ for the model on
${\mathbb{Z}}^{d}$). [4] only discusses spin $1/2$ but to get that equality
holds in (4.6), it suffices to get a MF lower bound on $T_{c}(\mu)$ and [4]
get that from Fröhlich-Simon-Spencer [8] whose argument works for any spin
(for many models of interest including those of the next two sections, there
is also a mean field upper bound on transition temperatures - see [24, Section
2.6] and [18, 25]). In event, we see that if (4.6) holds for all ferromagnetic
pair interactions, then the constant $\langle x^{2}\rangle_{\mu}$ is best
possible. A major theme of the rest of the paper is proving (4.6) in two
classes of models.

One of these is the spin $S$ measure and we want to end this section by noting
what we believe is the best prior lower bound on $T_{c}(\tilde{\mu}_{S})$. To
be explicit, for each value of $S=1/2,1,3/2,...$, consider the measure
$\tilde{\mu}_{S}$ which takes $2S+1$ values equally spaced between $-1$ and
$1$, each with weight $1/(2S+1)$. It is interesting to find the square,
$T_{-}(S)^{2}$, of the Wells $T_{-}$ associated to $\tilde{\mu}_{S}$ because
if $T_{c}(S)$ is the transition temperature for a model with apriori measure
$\tilde{\mu}_{S}$ and some fixed two point ferromagnetic interaction, then
Proposition 4.1 (and the easy $T_{+}(S)=1$) implies that

$T_{-}(S)^{2}T_{c}\left(\tfrac{1}{2}\right)\leq T_{c}(S)\leq
T_{c}\left(\tfrac{1}{2}\right)$ (4.8)

(we’ll prove in Section 6 that $\tilde{\mu}_{S}$ is canonical and compute
$\langle x^{2}\rangle_{\mu_{S}}$). So far as we know the best previous result
of this genre in the literature is due to Griffiths [12, eq.(4.23)] who proved
that

######  Theorem 4.2 (Griffiths [12]).

One has that

$\tfrac{1}{4}T_{c}\left(\tfrac{1}{2}\right)\leq T_{c}(S)\leq
T_{c}\left(\tfrac{1}{2}\right)$ (4.9)

######  Remarks.

1\. This is what Griffiths proves for $S$ an integer. For $2S=2k+1$ odd, he
proves the slightly strong result with $\tfrac{1}{4}$ replaced by
$(k+1/2k+1)^{2}$. Please note that what we call $T_{c}(S)$, Griffiths denotes
$T_{c}(2S)$, e.g. $T_{c}(\text{{classical Ising}})$ which we denote
$T_{c}\left(\tfrac{1}{2}\right)$, he denotes as $T_{c}(1)$.

2\. This paper of Griffiths [12] is best known for proving Lee-Yang and GKS
inequalities by realizing spin $S$ (normalized so the maximum value is $2S$, a
measure we call $\mu_{S}$) by $2S$ spins with values $\pm 1$ with finite
ferromagnetic couplings (given by Figs 2 and 3 in his paper) but he notes that
one can also realize them with $S$ (if $S$ is an integer or $S+\tfrac{1}{2}$
if that’s an integer) frozen together and then GKS implies that
$\tilde{\mu}_{S}$ Ising dominates $b_{T=1/2}$.

##  5\. Totally Anisotropic D-vector model

We turn next to one of the two new results on this subject. It involves the
interesting measure

$d\mu_{D}(x)=\left[\frac{\Gamma\left(\tfrac{D}{2}\right)}{\sqrt{\pi}\,\Gamma\left(\tfrac{D-1}{2}\right)}\right](1-x^{2})^{\tfrac{1}{2}(D-3)}\chi_{[-1,1]}(x)dx$
(5.1)

This is the distribution of $x_{1}$ if one looks at a $D$-component unit
vector, $\mathbf{x}=(x_{1},\dots,x_{D})$, distributed with the rotation
invariant probability measure on ${\mathbb{S}}^{D-1}$. Since, with respect to
this measure, all $x_{j}$ have the same distribution and
$\sum_{j=1}^{D}x_{j}^{2}=1$, we clearly have that

$\langle x^{2}\rangle_{D}=1/D$ (5.2)

######  Theorem 5.1.

$T_{-}(\mu_{D})$ is canonical, i.e. $T_{-}(\mu_{D})^{2}=1/D$, so, in
particular, (4.6) holds for $\mu_{D}$.

The result for $D=2$ is especially easy because, for all $m$,
$\langle(x^{2}-1/2)^{2m+1}\rangle_{D=2}=0$. To prove this, note that it is
equivalent to
$\langle(2x^{2}-1)^{2m+1}\rangle_{D=2}=\langle(x_{1}^{2}-x_{2}^{2})^{2m+1}\rangle_{\text{rotor}}=0$
which follows by $x_{1}\leftrightarrow x_{2}$. I note that this result for
$D=2$ is precisely the result that Aizenman and the second author say is in
Wells’ mystery preprint.

######  Lemma 5.2.

Let $\mu,\nu$ be two measures of compact support on $[0,\infty)$. Then

$\int f(x)\,d\mu(x)\leq\int f(x)\,d\nu(x)$ (5.3)

for all monotone functions with $f(0)=0$ if and only if

$\forall_{C>0}\,\mu([C,\infty))\leq\nu([C,\infty))$ (5.4)

###### Proof.

If (5.3) holds, we take $f$ to be the characteristic function of $[C,\infty)$
to get (5.4).

Now suppose that we have (5.4). By a simple approximation argument, it
suffices to prove (5.3) for $f$’s which are $C^{1}$ with $f(0)=0$. For such
$f$’s, we have that

$f(x)=\int_{0}^{\infty}f^{\prime}(C)\chi_{[C,\infty)}(x)\,dC$ (5.5)

by doing the integral. Thus

$\int f(x)\,d\mu(x)=\int_{0}^{\infty}f^{\prime}(C)\mu([C,\infty))\,dC$ (5.6)

$f$ monotone implies that $f^{\prime}\geq 0$, so that (5.4) implies (5.3). ∎

######  Lemma 5.3.

Let $a,b>0$ and suppose that $g$ is positive on $(-b,a)$ with

$\int_{-b}^{0}g(x)\,dx=\int_{0}^{a}g(x)\,dx;\qquad 0\leq y<b\Rightarrow
g(-y)\geq g(y)$ (5.7)

Let $f$ be an odd, monotone increasing, continuous function on $(-b,a)$. Then

$\int_{-b}^{a}f(x)g(x)\,dx\geq 0$ (5.8)

######  Remark.

The result is quite intuitive. The condition on $g$ says that the measure
$g(x)\,dx$ is concentrated on the right at larger $x$ than on the left, so
more concentrated where $|f|$ is larger.

###### Proof.

Define the measures on $[0,\infty)$:

$d\nu(x)=\chi_{[0,a]}(x)g(x)\,dx;\qquad d\mu(x)=\chi_{[0,b]}(x)g(-x)\,dx$
(5.9)

We claim that for all $C>0$, one has that

$\nu([0,C))\leq\mu([0,C))$ (5.10)

so, by (5.7), we have that (5.4) holds. Thus, since $f$ is monotone, by Lemma
5.2, we have that

$\int_{0}^{b}f(x)g(-x)\,dx\leq\int_{0}^{a}f(x)g(x)\,dx$ (5.11)

which is (5.8) since $f$ is odd. So we need only prove (5.10).

If $C\leq b$, (5.10) is immediate from the fact that for $x\geq 0$, $g(-x)\geq
g(x)$. By (5.7), if $C>b$, then

$\mu([0,C))=\mu([0,\infty))=\nu([0,\infty))\geq\nu([0,C))$ (5.12)

∎

###### Proof of Theorem 5.1.

As noted after the Theorem, $1/D$ is the second moment. Thus, by (3.12),
$T_{-}^{2}\leq 1/D$, so, by (3.3), we need only prove that for all
$m\in{\mathbb{N}}$, the have that

$\int_{-1}^{1}\left(x^{2}-\tfrac{1}{D}\right)^{2m+1}(1-x^{2})^{\tfrac{1}{2}(D-3)}\,dx\geq
0$ (5.13)

Note first that, since the integrand is even in $x$, we can integrate only
from $0$ to $1$ and then change variables from $x$ to $y=x^{2}-1/D$. One sees
that (5.13) is equivalent to

$\int_{-1/D}^{1-1/D}\frac{y^{2m+1}\left(1-\tfrac{1}{D}-y\right)^{\tfrac{1}{2}(D-3)}}{\left(y+\tfrac{1}{D}\right)^{1/2}}\,dy\geq
0$ (5.14)

If $D=2$, then the integral goes from $-1/2$ to $1/2$ and the integrand is
$y^{2m+1}(y^{2}-\tfrac{1}{4})^{-1/2}$ which is odd so the integral in (5.14)
is $0$ for all $m$, recovering what we saw above. If $D\geq 3$, define, for
$m\geq 1$

$g(y)=\frac{|y|\left(1-\tfrac{1}{D}-y\right)^{\tfrac{1}{2}(D-3)}}{\left(y+\tfrac{1}{D}\right)^{1/2}};\qquad
f(y)=\mathrm{sgn}(y)y^{2m}$ (5.15)

The fact that $1/D$ is the second moment of $d\mu_{D}$ implies the integral in
(5.13) vanishes if $m=0$ which means that $g$ obeys the first equation in
(5.7). Since $g(y)/|y|$ is monotone decreasing in $y$, the inequality in the
second half of (5.7) holds. $f$ is odd and monotone increasing, so (5.8)
implies (5.14). ∎

##  6\. Spin S

For each value of $S=1/2,1,3/2,...$, consider the measure $\tilde{\mu}_{S}$
which takes $2S+1$ values equally spaced between $-1$ and $1$, each with
weight $1/(2S+1)$. We will prove here that except for $S=1$, this measure is
canonical which will lead to improvements in the first inequality in (4.9) by
a factor of at least $\tfrac{4}{3}$ and which by discussion after (4.7) yields
optimal constants. We begin by computing $\langle
x^{2}\rangle_{\tilde{\mu}_{S}}$

######  Theorem 6.1.

We have that

$a_{S}\equiv\int x^{2}d\tilde{\mu}_{S}(x)=\frac{1}{3}\frac{S+1}{S}$ (6.1)

######  Remarks.

1\. Using $\sum_{j=1}^{S}j^{2}=\tfrac{S(S+1)(2S+1)}{6}$, one easily gets this
result for $S$ integral. One can also use this formula to get the result for
half odd integral $S$ by using the fact that this case can be rewritten as a
sum over odd integers between $1$ and $2S$ which one can realize as a sum over
all integers minus the sum over even integers. That calculation is awkward and
it is surprising it yields the same algebraic formula which is why we give the
slick uniform proof below.

2\. We will often scale the measure by a factor of $S$ so the $2S+1$ values
are equally spaced from $-S$ to $S$ yielding a measure $\mu_{S}$. If we
interpret $\sum_{j=-S}^{S}$ as the sum over the $-S,-S+1,\dots,S-1,S$ (the
usual meaning if $S$ is an integer but unusual if $S$ is half and odd
integer), then (6.1) is equivalent to

$A_{S}\equiv\int
x^{2}d\mu_{S}(x)=\frac{1}{2S+1}\sum_{-S}^{S}j^{2}=\frac{S(S+1)}{3}$ (6.2)

###### Proof.

Use $\text{\sout{\rm{Tr}}}(\cdot)$ for the normalized trace on finite
dimensional spaces, i.e. the average of diagonal elements or
$\text{\rm{Tr}}(\cdot)$ divided by the dimension. Consider what a physicist
would call a quantum spin, $\mathbf{\sigma}$, of spin $S$ and a mathematician
the generators of the irreducible representation of $SU(2)$ of dimension
$2S+1$. The operator $\sigma_{z}$ has eigenvalues $-S,-S+1,\dots,S-1,S$ and
the Casimir operator has the form:

$\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}=S(S+1){\boldsymbol{1}}$ (6.3)

Thus

$\displaystyle A_{S}$
$\displaystyle=\text{\sout{\rm{Tr}}}\left(\sigma_{z}^{2}\right)=\frac{1}{3}\text{\sout{\rm{Tr}}}(\left(\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}\right)$
$\displaystyle=\frac{1}{3}\text{\sout{\rm{Tr}}}\left(S(S+1){\boldsymbol{1}}\right)=\frac{S(S+1)}{3}$
(6.4)

by the symmetry of the three directions. This proves (6.2) ∎

We saw in Example 3.3 that for $S=1$ ($\lambda=2/3$), one has that
$T_{-}=\sqrt{\tfrac{1}{2}}<\sqrt{\tfrac{2}{3}}=\sqrt{a_{S=1}}$, so
$\tilde{\mu}_{S}$ is not canonical for that value of $S$. The main technical
result of this section, which we will prove in Appendix A, is

######  Theorem 6.2.

Let $S\neq 1$ be half a positive integer. Then

$\sum_{j=-S}^{S}(3j^{2}-S(S+1))^{2m+1}\geq 0$ (6.5)

We will also see that the opposite inequality is true when $S=1$. By Theorem
3.1, this implies that

######  Corollary 6.3.

For any $S\neq 1$, we have that $T_{-}$ is canonical for $\tilde{\mu}_{S}$.

which in turn, by (4.8) implies that

######  Corollary 6.4.

For $S\neq 1$, one has that

$T_{c}(S)\geq\left(\frac{1}{3}+\frac{1}{3S}\right)T_{c}\left(\tfrac{1}{2}\right)$
(6.6)

while for $S=1$ one has that

$T_{c}(1)\geq\frac{1}{2}T_{c}\left(\tfrac{1}{2}\right)$ (6.7)

This improves Griffiths result, (4.9) by a factor of at least $4/3$. To be
totally accurate, as we noted Griffith’s lower bound (as proven, not as
stated!) is slightly stronger than (4.9) when $S$ is half an odd integer. In
particular for $S=\tfrac{3}{2}$ what (4.9) has as $\tfrac{1}{4}$, he actually
proves is $\tfrac{4}{9}$ so our results are improvement but for this value
only by a factor of $5/4$ rather than $4/3$.

One final remark. In the notation used in (6), (6.5) is equivalent to

$\text{\rm{Tr}}\left((2\sigma_{z}^{2}-\sigma_{x}^{2}-\sigma_{y}^{2})^{2m+1}\right)\geq
0$ (6.8)

We wonder if that couldn’t be used for a matrix theoretic version of what we
prove in the Appendix.

##  Appendix A A Majorization Bound

In this appendix, we will prove a general set of inequalities that includes
(6.5). We will prove two theorems

######  Theorem A.1.

Fix an integer $N\geq 1$, a function, $\psi$ on $[0,1]$, which is non-
negative, continuous, strictly monotone increasing and convex and a function,
$\Phi$, on $[-\lVert\psi\rVert_{\infty},\lVert\psi\rVert_{\infty}]$ which is
continuous, odd and whose restriction to $[0,\lVert\psi\rVert]$ is convex. Let

$\overline{\psi}=(N+1)^{-1}\sum_{j=0}^{N}\psi\left(\tfrac{j}{N}\right)$ (A.1)

Then

$\sum_{j=1}^{N}\Phi\left(\psi\left(\tfrac{j}{N}\right)-\overline{\psi}\right)\geq
0$ (A.2)

######  Remark.

By translation and scaling, this result can easily be generalized. For
example, while stated for $N+1$ equally spaced points between $0$ and $1$, we
will apply it to $N+1$ odd integers stating at $1$. The map $k\mapsto(k-1)/2N$
maps those $N+1$ integers into the points of the theorem.

######  Theorem A.2.

For an integer $N\geq 2$ and an even, continuous, convex function, $\psi$ on
$[-1,1]$ and a function, $\Phi$, on
$[-\lVert\psi\rVert_{\infty},\lVert\psi\rVert_{\infty}]$ which is continuous,
odd and whose restriction to $[0,\lVert\psi\rVert]$ is convex. Let
$\overline{\psi}$ be given by

$\overline{\psi}=(2N+1)^{-1}\sum_{j=-N}^{N}\psi\left(\tfrac{j}{N}\right)$
(A.3)

Suppose that

$2\psi(1)+\psi(0)+2\psi\left(\tfrac{1}{N}\right)\geq 5\overline{\psi}$ (A.4)

If $N$ is odd, suppose that

$\psi\left(\tfrac{1}{2}+\tfrac{1}{2N}\right)\leq\overline{\psi}$ (A.5)

Then

$\sum_{j=-N}^{N}\Phi\left(\psi\left(\tfrac{j}{N}\right)-\overline{\psi}\right)\geq
0$ (A.6)

######  Remarks.

1\. We will see later (see (A.16)) that $2\psi(1)+2\psi(0)\geq
4\overline{\psi}$ which puts (A.4) in perspective.

2\. It might be true that this theorem holds without the need for the
condition (A.4) but it holds in the case we need so we didn’t try hard to
eliminate it. We will explain later why a naive extension of the proof of
Theorem A.1 doesn’t work in a simple example and led to the extra condition.
We do note that (A.4) is a restriction. If we normalize $\psi$ by
$\psi(0)=0,\psi(1)=1$, then in the limit as $N\to\infty$, (A.4) becomes

$\int_{0}^{1}\psi(x)\,dx\leq\tfrac{2}{5}$ (A.7)

which for $\psi(x)=|x|^{p}$ requires $p\geq\tfrac{3}{2}$ while convexity only
requires $p\geq 1$.

3\. On the other hand, (A.5), as we will see, is quite natural independent of
our method of proof.

One key idea behind the proofs is the theory of majorization. We let
${\mathbb{R}}^{n}_{+,\geq}$ denote the set of $n$-tuples of reals,
$\mathbf{x}=(x_{1},\dots,x_{n})$ with

$x_{1}\geq x_{2}\geq\dots x_{n}\geq 0$ (A.8)

If $\mathbf{x},\mathbf{y}\in{\mathbb{R}}^{n}_{+,\geq}$, we say that
$\mathbf{x}$ majorizes $\mathbf{y}$, written $\mathbf{x}\succ\mathbf{y}$ or
$\mathbf{y}\prec\mathbf{x}$ if an only if

$\sum_{j=1}^{n}x_{j}=\sum_{j=1}^{n}y_{j};\quad
S_{k}(\mathbf{x})\equiv\sum_{j=1}^{k}x_{j}\geq\sum_{j=1}^{k}y_{j},\,k=1,\dots,n-1$
(A.9)

which defines $S_{k}(\mathbf{x})$. A standard reference is Marshall-Olkin [17]
which has been called a love poem to majorization; other references are Hardy-
Littlewood-Pólya [10] and Simon [23, Chapters 14-15]. We will rely on the
following aspect of the theory:

######  Theorem A.3 (Karamata’s Inequality).

Let $\mathbf{x},\mathbf{y}\in{\mathbb{R}}^{n}_{+,\geq}$ with
$\mathbf{x}\succ\mathbf{y}$ and let $\varphi$ be an arbitrary continuous
convex function on $[0,x_{1}]$. Then

$\sum_{j=1}^{n}\varphi(x_{j})\geq\sum_{j=1}^{n}\varphi(y_{j})$ (A.10)

######  Remarks.

1\. Even though this is widely referred to as Karamata’s inequality (e.g.
Cvetkovski[7] or Wikipedia[27]) after Karamata’s 1932 paper [15], it or
theorems that imply it appear in a 1923 paper of Schur [19] and a 1929 paper
of Hardy-Littlewood-Pólya [9]. That said, we note that [9] doesn’t have a
proof which may not have appeared until [10] in 1934 and that Karamata proved
a converse, namely, if $\mathbf{x},\mathbf{y}\in{\mathbb{R}}^{n}_{+,\geq}$ and
(A.10) holds for all convex $\varphi$, then $\mathbf{x}\succ\mathbf{y}$.

2\. The idea of one proof is quite simple (for details, see, for example,
Simon [22, Theroem 1.9] or Simon [23, Theroem 15.5]): by slicing with specific
hyperplanes, one proves that $\mathbf{y}$ is in the convex hull in
${\mathbb{R}}^{n}$ of the (at most) $n!$ points obtained from $\mathbf{x}$ by
permuting the coordinates and then one notes the function
$\mathbf{w}\mapsto\sum_{j=1}^{n}\varphi(w_{j})$ is convex and permutation
symmetric.

In the case of Theorem A.1, we will prove that $\mathbf{x}\succ\mathbf{y}$
using a simple observation:

######  Proposition A.4.

Suppose that $\mathbf{x},\mathbf{y}\in{\mathbb{R}}^{n}_{+,\geq}$ with
$\sum_{j=1}^{n}x_{j}=\sum_{j=1}^{n}y_{j}$ and that for some $\ell\in
2,\dots,n-1$, one has that

$j<\ell\Rightarrow x_{j}>y_{j}\qquad\qquad j\geq\ell\Rightarrow x_{j}\leq
y_{j}$ (A.11)

Then $\mathbf{x}\succ\mathbf{y}$.

###### Proof.

If $k<\ell$, it is immediate that $\sum_{j=1}^{k}x_{j}\geq\sum_{j=1}^{k}y_{j}$
and similarly, it is immediate that if $k\geq\ell$, then
$\sum_{j=k}^{n}x_{j}\leq\sum_{j=k}^{\ell}y_{j}$. Subtracting this from
$\sum_{j=1}^{n}x_{j}=\sum_{j=1}^{n}y_{j}$, we see that also for $k\geq\ell$,
one has that $\sum_{j=1}^{k}x_{j}\geq\sum_{j=1}^{k}y_{j}$. ∎

We need two preliminaries for the proof of Theorem A.1:

######  Lemma A.5.

Let $\psi$ be a convex function on $[0,1]$ and suppose that

$0\leq\tilde{b}\equiv 2c-b<\tilde{a}\equiv 2c-a\leq c\leq a<b\leq 1$ (A.12)

Then

$\tfrac{1}{2}(\psi(b)+\psi(\tilde{b}))\geq\tfrac{1}{2}(\psi(a)+\psi(\tilde{a}))\geq\psi(c)$
(A.13)

Moreover, the first inequality is strict unless $\psi^{\prime}(s)$ is constant
on $(\tilde{a},a)$.

###### Proof.

If one takes $a=c$ and then replaces $b$ by $a$, the first inequality becomes
the second so it suffices to prove the first one. Without loss (by translation
and scaling) we can take $c=\tfrac{1}{2},b=1$ so that $\tilde{b}=1$ and
$\tilde{a}=1-a$. By the fundamental theorem of calculus (a general convex
function is not $C^{1}$ but it is differentiable with the possible exception
of a countable set and the fundamental theorem of calculus holds; see Simon
[23, Theorem 1.28])

$\tfrac{1}{2}(\psi(1)+\psi(0))-\tfrac{1}{2}(\psi(a)+\psi(\tilde{a}))=\tfrac{1}{2}\int_{a}^{1}[\psi^{\prime}(s)-\psi^{\prime}(1-s)]\,ds$
(A.14)

By convexity, the integrand is non-negative so we have proven (A.13). Moreover
if $\psi^{\prime}(s)$ is not constant on $(1-a,a)$, then the integral is
strictly positive. ∎

######  Proposition A.6.

Let $\psi$, $\overline{\psi}$ and $N$ be as in Theorem A.1. Then

$n\equiv\\#\\{j\,\mid\,\psi\left(\tfrac{j}{N}\right)\leq\overline{\psi}\\}\geq(N+1)/2$
(A.15)

and

$\psi\left(\tfrac{1}{2}\right)\leq\overline{\psi}\leq\tfrac{1}{2}(\psi(0)+\psi(1))$
(A.16)

Moreover, the inequalities in (A.16) are strict if $N\geq 2$ and $\psi$ is not
an affine function on $[0,1]$ (i.e. $\psi^{\prime}$ is not constant).

###### Proof.

For any $j=0,1,\dots,N$, (A.13) implies that

$\psi\left(\tfrac{1}{2}\right)\leq\tfrac{1}{2}\left(\psi\left(\tfrac{j}{N}\right)+\psi\left(1-\tfrac{j}{N}\right)\right)\leq\tfrac{1}{2}(\psi(0)+\psi(1))$
(A.17)

Averaging over $j$ yields (A.16). If $\psi$ is not affine on $[0,1]$, then the
second inequality is strict for $1\leq j\leq N-1$ so the second inequality in
(A.16) is strict. Since
$\psi\left(\tfrac{1}{2}\right)<\tfrac{1}{2}(\psi(0)+\psi(1))$ if $\psi$ is not
affine, we see that in the case the first inequality is always strict.

Since $\psi$ is strictly monotone, the first inequality in (A.16) implies the
unique $x\in[0,1]$ with $\psi(x)=\overline{\psi}$ has $x\geq\tfrac{1}{2}$.
This implies that $n=\\#\\{j\,\mid\,\tfrac{j}{N}\leq
x\\}\geq\\#\\{j\,\mid\,\tfrac{j}{N}\leq\tfrac{1}{2}\\}\geq(N+1)/2.$ ∎

###### Proof of Theorem A.1.

Let $q=N+1-n\leq n$ by (A.15). Define

$y_{j}=\overline{\psi}-\psi\left(\tfrac{j}{N}\right)\qquad j=0,\dots,n-1$
(A.18)
$x_{j}=\left\\{\begin{array}[]{ll}\psi\left(\tfrac{N+1-j}{N}\right)-\overline{\psi}&\hbox{
if }j=1,\dots,q\\\ 0,&\hbox{ if }j\geq q\end{array}\right.$ (A.19)

Since $\psi$ is monotone and $n$ is defined by (A.15), we have that
$\mathbf{x},\mathbf{y}\in{\mathbb{R}}^{n}_{+,\geq}$. By the definition of
$\overline{\psi}$, we have that

$\sum_{j=1}^{n}x_{j}=\sum_{j=1}^{n}y_{j}$ (A.20)

If $N=1$ or $\psi$ is affine on $[0,1]$, it is easy to see that $x_{j}=y_{j}$
for all $j$, so, since $\Phi$ is odd, we have that (A.8) holds. Thus
henceforth we will suppose that $N\geq 2$ and $\psi$ is not an affine function
on $[0,1]$, so, in particular, the inequalities in (A.16) are strict.

Note next that because $\psi$ is assumed convex, we have that

$m<p\Rightarrow\psi\left(\tfrac{m+1}{N}\right)-\psi\left(\tfrac{m}{N}\right)\leq\psi\left(\tfrac{p+1}{N}\right)-\psi\left(\tfrac{p}{N}\right)$
(A.21)

By the strict form of (A.16), $x_{1}>y_{1}$. Because of (A.20), there must be
a first $\ell$ so that $x_{\ell}\leq y_{\ell}$. We claim that if $\ell<n$,
then $x_{\ell+1}\leq y_{\ell+1}$. If $\ell+1>q$, then $x_{\ell+1}=0$ and the
required inequality is immediate. If $\ell+1\leq q$, then (A.21) implies that
$x_{\ell}-x_{\ell+1}\geq y_{\ell}-y_{\ell+1}$. Subtracting this from
$x_{\ell}\leq y_{\ell}$ proves that $x_{\ell+1}\leq y_{\ell+1}$. Repeating
this argument, proves that for all $j\geq\ell$ we have that $x_{j}\leq y_{j}$.
Thus by Proposition A.4, $\mathbf{x}\succ\mathbf{y}$.

By Karamata’s inequality, (A.10), we conclude that
$\sum_{j=1}^{n}\Phi(x_{j})-\Phi(y_{j})\geq 0$. Since $\Phi$ is odd and
$\Phi(0)=0$, this is equivalent to (A.8). ∎

######  Example A.7.

To understand why we need the extra condition (A.4) in Theorem A.2, we
consider $\mu_{S}$ for $S=6$, i.e. $13$ pure points with weight $1/13$ at
$0,\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6$. By (6.2), the average of the square
is $A_{6}=14$. The values of $j^{2}$ are
$j^{2}=0,1,1,4,4,9,9,16,16,25,25,36,36$ so $n=7$ values are less than $A_{6}$
and one sees that (ignore $\mathbf{w}$ for now)

$\displaystyle\mathbf{x}$ $\displaystyle=22,22,11,11,\,\;2,2,0$
$\displaystyle\mathbf{y}$ $\displaystyle=14,13,13,10,10,5,5$
$\displaystyle\mathbf{w}$ $\displaystyle=22,22,\,\;0,11,11,2,2$ (A.22)

One can verify that $\mathbf{x}\succ\mathbf{y}$ by hand (and, below, we will
prove the result for all $S\geq 2$) but one can’t use Proposition A.4 as we
did in our proof of Theorem A.1 for $x_{j}-y_{j}$ shifts signs three times
instead of one time. The problem is that the components of $\mathbf{x}$ and
$\mathbf{y}$ are paired but shifted.

Look at $\mathbf{w}$ which we get by moving the $0$ from position $7$ to
position $3$. One can handle the first three partial sums by seeing that
$22+22\geq 14+13+13$ and the remaining partial sums by noting that there is
only one sign shift after the third place and use Proposition A.4 to prove the
partial sums of $\mathbf{w}$ dominate those of $\mathbf{y}$ and note it is
trivial that partial sums of $\mathbf{x}$ dominate those of $\mathbf{w}$. The
key is that by moving the $0$, the pairs are no longer shifted.

###### Proof of Theorem A.2.

Notice that $2N+1$ is odd, so either $\mathbf{x}$ or $\mathbf{y}$ needs to
have a zero added. To be sure that it is added to the $\mathbf{x}$, we need
the analog of (A.15), namely that

$n\equiv\\#\\{j\,\mid\,\psi\left(\tfrac{j}{N}\right)\leq\overline{\psi}\\}\geq
N+1$ (A.23)

If $N=2k$ is even, the $N+1$ values of $j$ among the $2N+1$ overall values
with the smallest values of $\psi\left(\tfrac{j}{N}\right)$ are $0,\pm 1,\pm
k$, so (A.23) follows from the first inequality in (A.16). If $N=2k+1$ is odd,
we can’t take the max $j$ to be $k$ since that only yields $2k+1<N+1$ values
and therefore we need to go up to $j=k+1$ yielding $N+2$ values (leaving over
$N-1$), so we need (A.5) to hold. In that case, there are at least three more
values $\leq\overline{\psi}$ than $>\overline{\psi}$! (This has to be because
$n$ is odd and $N-n$ even and we require that $n>N-n$.)

Define $\mathbf{x}$ to be the $2N+1-n$ values of $\psi-\overline{\psi}$ larger
than $0$ written in decreasing order plus $2n-2N+1$ zero values at the end and
$\mathbf{y}$ the $n$ non-negative values of $\overline{\psi}-\psi$ so
$\mathbf{x},\mathbf{y}\in{\mathbb{R}}^{n}$. As in the proof of Theorem A.1, if
we prove that $\mathbf{x}\succ\mathbf{y}$, then (A.6) follows from Karamata’s
inequality.

For $N\geq 2$ (so $n\geq 3$), define $\mathbf{w}\in{\mathbb{R}}^{n}_{+,\geq}$
by

$w_{j}=\left\\{\begin{array}[]{ll}x_{j},&\hbox{ if }j=1,2\\\ 0,&\hbox{ if
}j=3\\\ x_{j-1},&\hbox{ if }j\geq 3\end{array}\right.$ (A.24)

We claim first that

$S_{n}(\mathbf{x})=S_{n}(\mathbf{w});\qquad S_{j}(\mathbf{x})\geq
S_{j}(\mathbf{w}),\,j=1,\dots,n-1$ (A.25)

(we do not write this as $\mathbf{x}\succ\mathbf{w}$, first because
$\mathbf{w}\notin{\mathbb{R}}^{n}_{+,\geq}$ and we’ve only defined
majorization for such sequences and also because it is usual to extend $\succ$
to ${\mathbb{R}}^{n}_{+}$ by demanding the relations for the decreasing
rearrangements rather than by (A.25)). This follows because $\mathbf{w}$ is a
rearrangement of $\mathbf{x}$ and the decreasing sequence,
$S_{j}(\mathbf{x})$, is maximal among those sums for any arrangement or also,
more directly, by noting that $S_{j}(\mathbf{w})=S_{j}(\mathbf{x})$ for
$j=1,2,n$ and $S_{j}(\mathbf{w})=S_{j-1}(\mathbf{x})$ for $j\geq 3$.

Thus, if we prove that

$S_{n}(\mathbf{w})=S_{n}(\mathbf{y});\qquad S_{j}(\mathbf{w})\geq
S_{j}(\mathbf{y}),\,j=1,\dots,n-1$ (A.26)

then $\mathbf{x}\succ\mathbf{y}$ and the proof is done.

(A.26) for $j=n$ follows from the definition of $\overline{\psi}$. For
$j=1,2$, we note that by the second inequality of (A.16), we have that
$w_{1}=x_{1}\geq y_{1}$ and then that
$S_{2}(\mathbf{w})=S_{2}(\mathbf{x})=2x_{1}\geq 2y_{1}\geq y_{1}+y_{2}$. For
$j=3$, the required inequality is the hypothesis (A.4).

Because of the equality for $j=n$ and inequality for $j=3$, there must be a
first $\ell\geq 4$ with $x_{\ell}\leq y_{\ell}$. We claim either
$x_{\ell+1}=0$ or $x_{\ell}-x_{\ell+1}\geq y_{\ell}-y_{\ell+1}$ because if
$x_{\ell+1}>0$ and $\ell$ is odd, we have that $x_{\ell}=x_{\ell+1}$ and
$y_{\ell}=y_{\ell+1}$ or $\ell$ is even and we can use (A.21) to prove the
inequality as in the proof of Theorem A.1. By induction we see that for
$j\geq\ell$, we have that $x_{j}\leq y_{j}$. By following the argument used to
prove Proposition A.4 we conclude that for all $j\geq 4$, we have (A.26). ∎

###### Proof of Theorem 6.2.

If $S$ is half an odd integer, we can use Theorem A.1 with $N=S-\tfrac{1}{2}$
and $\psi(x)=\left(\tfrac{1}{2}+Nx\right)^{2}$ which is non-negative, strictly
monotone and convex and $\Psi(y)=y^{2m+1}$.

If $S$ is an integer, we will use Theorem A.2 with $N=S$, $\psi(x)=(Sx)^{2}$
and $\Psi(y)=y^{2m+1}$. We need to check (A.4), which says that

$2S^{2}+2\geq 5A_{S}=\tfrac{5}{3}S(S+1)$ (A.27)

and (A.5) which says that for $S\geq 3$ odd, one has that

$S^{2}\left(\tfrac{1}{2}+\tfrac{1}{2S}\right)^{2}\leq A_{S}=\tfrac{S(S+1)}{3}$
(A.28)

(A.27) is equivalent to

$0\leq S^{2}-5S+6=(S-2)(S-3)$ (A.29)

which holds for all integral $S$ (since the polynomial $x^{2}-5x+6$ is
negative precisely for $x\in(2,3)$). (A.28) is equivalent to $3S^{2}+6S+3\leq
4S^{2}+4S$ or

$0\leq S^{2}-2S-3=(S-3)(S+1)$ (A.30)

which holds for all $S\geq 3$ as required. ∎

We note that when $S=1$ with the normalization used in this proof, one has
that the values of $j^{2}$ are $0,1,1$ so $A_{1}=\tfrac{1}{3}$ and
$\mathbf{x}=(\tfrac{1}{3},\tfrac{1}{3})$ while $y=(\tfrac{2}{3},0)$ so
$\mathbf{x}\prec\mathbf{y}$ and for all $m$, (6.5) holds with the opposite
sign!

As we mentioned, we would guess that results like Theorem A.2 hold without
(A.4). To be explicit, consider the an analog of (6.5) with $3j^{2}$ replace
by $|j|^{p}$ and $3A_{S}=S(S+1)$ replaced by the suitable average. We know
this analog is valid for any $p>1$ if $S$ is half an odd integer but because
(A.4) fails when $p<3/2$ and $S$ is large we don’t know the answer to

Question 2 Does the analog of (6.5) hold for $S\neq 1$ integral when $3j^{2}$
is replaced by $|j|^{p}$ for any $p>1$ and with $3A_{S}=S(S+1)$ replaced by
the suitable average.

This is an explicit example for what we hope might be a general result.

## References

  * [1]
  * [2] Aizenman, M., and Simon, B.,  A comparison of plane rotor and Ising models, Phys. Lett. A76 (1980), 281–282.
  * [3] van Beijeren, H., and Sylvester, G. S,. Phase transitions for continuous-spin Ising ferromagnets, J. Func. Anal. 28 (1978), 145–167.
  * [4] Bricmont, J., Kesten, H., Lebowitz, J. L. and Schonmann, R. H., A note on the Ising model in high dimensions, Comm. Math. Phys. 122 (1989), 597–607.
  * [5] Bricmont, J., Lebowitz, J. L., and Pfister, C. E., Periodic Gibbs states of ferromagnetic spin systems, J. Stat. Phys. 24 (1981), 269–277.
  * [6] Chebyshev, P. L., On approximate expressions of some integrals in terms of others, taken within the same limits, Proc. Math. Soc. Kharkov 2 (1882), 93–98.
  * [7] Cvetkovski, Z., Inequalities: Theorems, Techniques and Selected Problems, Springer, 2012.
  * [8] Fröhlich, J., Simon, B., and Spencer, T., Infrared bounds, phase transitions and continuous symmetry breaking, Comm. Math. Phys. 50 1976, 79–95.
  * [9] Hardy, G. H., Littlewood, J. E., and Pólya, G., Some simple inequalities satisfied by convex functions, Messenger of Math. 58 (1929), 145–152.
  * [10] Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities, Cambridge University Press, Cambridge, 1934.
  * [11] Ginibre, J., General formulation of Griffths’ inequalities, Comm. Math. Phys. 16 (1970), 310–328.
  * [12] Griffiths, R. B., Rigorous results for Ising ferromagnets of arbitrary spin, J. Math. Phys. 10 (1969), 1559–1565.
  * [13] Griffiths, R. B., and Simon, B., The $(\varphi^{4})_{2}$ field theory as a classical Ising model, Comm. Math. Phys. 33 (1973), 145–164.
  * [14] Guerra, F., Rosen, L., and Simon, B., The $P(\phi)_{2}$ Euclidean quantum field theory as classical statistical mechanics, I, II, Ann. of Math. 101 (1975), 111–259.
  * [15] Karamata, J., Sur une inégalité rélative aux fonctions convexes, Publ. Math. Univ. Belgrade 1 (1932), 145–148.
  * [16] Lieb, E. H., The classical limit of quantum spin systems, Comm. Math. Phys. 31 (1973), 327–340.
  * [17] Marshall, A. W., and Olkin, I., Inequalities: Theory of Majorization and Its Applications, Academic Press, New York- London, 1979 (second edition with Arnold, B. C. in 2009)
  * [18] Pearce, P. A., Mean-field bounds on the magnetization for ferromagnetic spin models, J. Stat. Phys. 25 (1981), 309–320.
  * [19] Schur, I., Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsber. Berlin Math. Gesellschaft 22 (1923), 9–20.
  * [20] Simon, B., The classical limit of quantum partition functions, Comm. Math. Phys. 71 (1980), 247-276.
  * [21] Simon, B., The Statistical Mechanics of Lattice Gases, Princeton University Press, 1993.
  * [22] Simon, B. Trace Ideals and Their Applications, second edition, Mathematical Surveys and Monographs 120, American Mathematical Society, 2005 (first edition, 1979).
  * [23] Simon, B., Convexity: An Analytic Viewpoint, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011.
  * [24] Simon, B., Phase Transitions in the Theory of Lattice Gases, Cambridge University Press, in preparation.
  * [25] Tasaki, H., and Hara, T., Mean field bound and GHS inequality, J. Stat. Phys. 35 (1984), 99–107.
  * [26] Wells, D. R., Some moment inequalities and a result on multivariate unimodality, (unpublished) Indiana University Ph. D. Thesis, 1977
  * [27] Wikipedia article, Karamata’s Inequality,
$\underline{\text{https://en.wikipedia.org/wiki/Karamata}\%27s\\_\text{inequality}}$.


# A Standard Law for the Equatorward Drift of the Sunspot Zones

D. H. Hathaway1 1 NASA Marshall Space Flight Center, Huntsville, AL 35812 USA
email<EMAIL_ADDRESS>
###### Abstract

The latitudinal location of the sunspot zones in each hemisphere is determined
by calculating the centroid position of sunspot areas for each solar rotation
from May 1874 to June 2011. When these centroid positions are plotted and
analyzed as functions of time from each sunspot cycle maximum there appears to
be systematic differences in the positions and equatorward drift rates as a
function of sunspot cycle amplitude. If, instead, these centroid positions are
plotted and analyzed as functions of time from each sunspot cycle minimum then
most of the differences in the positions and equatorward drift rates
disappear. The differences that remain disappear entirely if curve fitting is
used to determine the starting times (which vary by as much as 8 months from
the times of minima). The sunspot zone latitudes and equatorward drift
measured relative to this starting time follow a standard path for all cycles
with no dependence upon cycle strength or hemispheric dominance. Although
Cycle 23 was peculiar in its length and the strength of the polar fields it
produced, it too shows no significant variation from this standard. This
standard law, and the lack of variation with sunspot cycle characteristics, is
consistent with Dynamo Wave mechanisms but not consistent with current Flux
Transport Dynamo models for the equatorward drift of the sunspot zones.

###### keywords:

Solar Cycle, Observations; Sunspots, Statistics; Sunspots, Velocity

##  1 Introduction

The equatorward drift of the sunspot zones is now a well known characteristic
of the sunspot cycle. While Carrington58 noted the disappearance of low
latitude spots followed by the appearance of spots confined to mid-latitudes
during the transition from Cycle 9 to Cycle 10, and Sporer80 calculated and
plotted the equatorward drift of sunspot zones over Cycles 10 and 11, the very
existence of the sunspot zones was still in question decades later [Maunder
(1903)]. The publication of the “Butterfly Diagram” by Maunder04 laid this
controversy to rest and revealed a key aspect of the sunspot cycle – sunspots
appear in zones on either side of the equator that drift toward the equator as
each sunspot cycle progresses.

Cycle-to-cycle variations in the sunspot latitudes have been noted previously.
Becker54 and Waldmeier55 both noted that, at maximum, the sunspot zones are at
higher latitudes in the larger sunspot cycles. More recently, Hathaway_etal03
found an anti-correlation between the equatorward drift rate and cycle period
and suggested that this was evidence in support of flux transport dynamos
[Nandy & Choudhuri (2002)]. However, Hathaway10 noted that all these results
are largely due to the fact that larger sunspot cycles reach maximum sooner
than smaller sunspot cycles and that the drift rate is faster in the earlier
part of both small and large cycles. Nonetheless, Hathaway10 did find that the
sunspot zones appeared at slightly higher latitudes (with slightly higher
drift rates) in the larger sunspot cycles when comparisons were made relative
to the time of sunspot cycle minimum.

The equatorward drift of the sunspot zones is a key characteristic of the
sunspot cycle. It must be reproduced in viable models for the Sun’s magnetic
dynamo and can be used to discriminate between the various models.

In the Babcock61 and Leighton69 dynamo models the latitudinal positions of the
sunspot zones are determined by the latitudes where the differential rotation
and initial magnetic fields produce magnetic fields strong enough to make
sunspots. This “critical” latitude moves equatorward from the position of
strongest latitudinal shear as the cycle progresses. The initial strength of
the magnetic field in these models is determined by the polar field strength
at cycle minimum so we might expect a delayed start for cycles starting with
weak polar fields and the equatorward propagation might depend on both the
differential rotation profile (which doesn’t vary substantially) and the
initial polar fields (which do vary substantially).

In a number of dynamo models (both kinematic and magnetohydrodynamic) the
equatorward drift of the sunspot zones is produced by a “Dynamo Wave” (cf.
Yoshimura75) which propagates along iso-rotation surfaces at a rate and
direction given by the product of the shear in the differential rotation and
the kinetic helicity in the fluid motions. In these models the equatorward
propagation is a function of the differential rotation profile and the profile
of kinetic helicity - both of which don’t vary substantially.

In flux transport dynamo models (cf. NandyChoudhuri02) the equatorward drift
is produced primarily by the equatorward return flow of a proposed deep
meridional circulation. In these models, variations in the meridional flow
speed (which does vary substantially with cycle amplitude and duration in
these models) should be observed as variations in the equatorward drift rate
of the sunspot zones.

Here we reexamine the latitudes of the sunspot zones and find that cycle-to-
cycle and hemispheric variations vanish when time is measured relative to a
cycle starting time derived from fitting the monthly sunspot numbers in each
cycle to a functional form for the cycle shape.

##  2 The Sunspot Zones

Sunspot group positions and areas have been measured daily since May 1874. The
Royal Observatory Greenwich carried out the earlier observations using a small
network of solar observatories from May 1874 to December 1976. The United
States Air Force and National Oceanic and Atmospheric Administration continued
to acquire similar observations from a somewhat larger network starting in
January 1977. We calculate the average daily sunspot area over each Carrington
rotation for 50 equal area latitude bins (equi-spaced in $\sin\lambda$ where
$\lambda$ is the latitude). The sunspot zones are clearly evident in the
resulting Butterfly Diagram - Figure 1.

Figure 1.: Sunspot areas as functions of sin(latitude) and time for each
Carrington rotation from 1880 to 2010. These data include four small cycles
(Cycles 12, 13, 14, and 16), four average cycles (Cycles 15,17, 18, and 20),
and four large cycles (Cycles 19, 21, 22, and 23).

We divide the data into separate sunspot cycles by attributing low-latitude
groups to the earlier cycle and high-latitude groups to the later cycle when
the cycles overlap at minima. We further divide the data by hemisphere and
then calculate the latitude, $\bar{\lambda}$, of the centroid of sunspot area
for each hemisphere for each rotation of each sunspot cycle using

$\bar{\lambda}=\sum{A(\lambda_{i})\lambda_{i}}/\sum{A(\lambda_{i})}$ (1)

where $A(\lambda_{i})$ is the average daily sunspot area in the latitude bin
centered on latitude $\lambda_{i}$ and the sums are over the 25 latitude bins
for a given hemisphere and Carrington rotation. These centroid positions then
provide the sunspot zone latitudes and drift rates for each hemisphere as a
function of time through each cycle.

##  3 The Sunspot Zone Equatorward Drift

Previous work on cycle-to-cycle variations in the positions and drift rates of
the sunspot zones [Becker (1954), Waldmeier (1955), Hathaway et al. (2003)]
made those measurements relative to the sunspot cycle maxima. The centroid
position data are plotted as functions of time from cycle maxima in Figure 2.
The data encompass 12 sunspot cycles which, fortuitously, include four cycles
much smaller than average (Cycles 12, 13, 14, and 16 with smoothed sunspot
cycle maxima below 90), four cycles much larger than average (Cycles 18, 19,
21, and 22 with smoothed sunspot cycle maxima above 150), and four cycles
close to the average (Cycles 15, 17, 20, and 23).

Figure 2 illustrates why the earlier studies concluded that larger cycles tend
to have sunspot zones at higher latitudes. The centroid positions for the
large cycles (in red) are clearly at higher latitudes than those for the
medium cycles which, in turn, are at higher latitudes than those for the small
cycles. While this conclusion is technically correct, it is somewhat
misleading since large cycles reach their maxima sooner than small cycles (the
“Waldmeier Effect” Waldmeier35 and Hathaway10) and the sunspot zones are
always at higher latitude earlier in each cycle.

Figure 2.: The centroid (area weighted) positions of the sunspot zones in each
hemisphere for each solar rotation are plotted as functions of time from each
sunspot cycle maximum. The individual data points are shown in the upper
panel. The size of the symbol varies with the average daily sunspot area for
each solar rotation and hemisphere. The color of the symbol varies with the
amplitude of the sunspot cycle associated with the data. The average centroid
positions of the sunspot zones for small (blue) medium (green) and large (red)
cycles plotted at 6-month intervals in time from sunspot cycle maximum are
shown in the lower panel.

In Figure 3 the centroid positions are plotted as functions of time from
sunspot cycle minima. Since large cycles reach maximum earlier than small
cycles, the data points for the large cycles are shifted to the left (closer
to minimum) relative to the medium and small cycles. The size of the shift is
different for each cycle depending on the dates of minimum and maximum.
Comparing Figures 2 and 3 shows that: 1) the latitudinal scatter is smaller in
Figure 3 than in Figure 2 and; 2) the differences in the centroid positions
for the different cycle amplitudes are diminished in Figure 3. This suggests
that there is a more general, cycle amplitude independent, law for the
latitudes (and consequently latitudinal drift rates) of the sunspot zones. A
slight additional shift in the adopted times for sunspot cycle minima (with
earlier times for small cycles) would appear to further diminish any cycle
amplitude differences.

Figure 3.: The centroid positions of the sunspot zones in each hemisphere for
each solar rotation are plotted as functions of time from sunspot cycle
minimum with the same method as in Figure 2.

Determinations of the dates of sunspot cycle minima are not well defined. Many
investigators simply take the date of minimum in some smoothed sunspot cycle
index (e. g. sunspot number, sunspot area, 10.7 cm radio flux). Unfortunately,
this can give dates that are clearly not representative of the actual cycle
minima. This problem led earlier investigators to define the date of minimum
as some (undefined) average of the dates of: 1) minimum in the monthly sunspot
number; 2) minimum in the smoothed monthly sunspot number; 3) maximum in the
number of spotless days per month; 4) predominance of new cycle sunspot groups
over old cycle sunspot groups [Waldmeier (1961), McKinnon (1987), Harvey &
White (1999)]. Even neglecting the fact the the nature of the average is not
defined, it is clear from the published dates for previous cycle minima that
the first criterium is never used (probably due to the wide scatter it gives)
and that even the simple average of the remaining criteria doesn’t give the
published dates [Hathaway (2010)].

An alternative to using this definition for the dates of sunspot cycle minima
is to use curve fitting to either the initial rise of activity or to the
complete sunspot cycle. Curve fitting is less sensitive to the noise
associated minimum cycle behavior (e.g. discretized data and missing data from
the unseen hemisphere). Hathaway_etal94 described earlier attempts at fitting
solar cycle activity levels (monthly sunspot numbers) to parameterized
functions and arrived at a function of just two parameters (cycle starting
time $t_{0}$ and cycle amplitude $R_{max}$) as the most useful function for
characterizing and predicting solar cycle behavior. This function:

$F(t;t_{0},R_{max})=R_{max}\ 2({t-t_{0}\over b})^{3}/\left[exp({t-t_{0}\over
b})^{2}-0.71\right]$ (2)

is a skewed Gaussian with an initial rise that follows a cubic in time from
the starting time (measured in months). The width parameter, $b$, is a
function of cycle amplitude $R_{max}$ that mimics the “Waldmeier Effect.” This
function is

$b(R_{max})=27.12+25.15(100/R_{max})^{1/4}$ (3)

Fitting $F(t;t_{0},R_{max})$ to the monthly averages of the daily
International Sunspot Numbers using the Levenberg-Marquardt method [Press et
al. (1986)] gives the amplitudes and starting times given by Hathaway_etal94
and reproduced in Table 1 with the addition of results for Cycle 23.

On average the small cycles have starting times about 7 months earlier than
minimum while medium cycles and large cycles have starting times about equal
to minimum. However, since minimum is determined by the behavior of both the
old and the new cycles, there are substantial differences between the dates of
minima and the starting times even among the medium and large cycles. For
example, Cycles 21 and 22 were both large but the minimum was 3 months earlier
than the starting time in Cycle 21 and 4 months later in Cycle 22. This is
illustrated in Figure 4.

Figure 4.: Monthly sunspot numbers for the Cycle 20/21 minimum (top) and Cycle
21/22 minimum (bottom). The curves fit to each cycle are shown with the
colored lines with the sum of both contributions indicated by the dashed black
line. Dates of minima and starting times are indicated to illustrate the
differences.

Measuring the time through each cycle relative to these starting times (rather
than minimum or maximum) removes the scatter and cycle amplitude dependence on
the centroid positions as shown in Figure 5.

Figure 5.: The centroid positions of the sunspot zones in each hemisphere for each solar rotation are plotted as functions of time from sunspot cycle start as determined by fitting a parameterized function to each cycle. The individual data points are shown in the upper panel. The average centroid positions of the sunspot zones for small (blue) medium (green) and large (red) cycles plotted at 6-month intervals in time from sunspot cycle start are shown in the lower panel. The average centroid positions for all of the data are shown with $2\sigma$ error bars. All three curves fall within the $2\sigma$ errors, criss-crossing each other along a common, standard trajectory given by the exponential fit in Equation 4 (dashed line) Table 1.: Sunspot cycle number, amplitude, minimum date, starting date, difference (starting date - minimum date in months), and dominant hemisphere. Cycle | Amplitude | Min | $t_{0}$ | $\Delta$ | Hemisphere
---|---|---|---|---|---
12 | 75 (small) | 1878/12 | 1878/05 | -7 | South
13 | 88 (small) | 1890/01 | 1889/05 | -8 | South
14 | 64 (small) | 1901/12 | 1901/08 | -4 | Balanced
15 | 105 (medium) | 1913/06 | 1913/03 | -3 | North
16 | 78 (small) | 1923/09 | 1923/02 | -7 | North
17 | 119 (medium) | 1933/10 | 1933/11 | +1 | Balanced
18 | 151 (large) | 1944/02 | 1944/03 | +1 | South
19 | 201 (large) | 1954/04 | 1954/04 | 0 | North
20 | 111 (medium) | 1964/10 | 1964/11 | +1 | North
21 | 164 (large) | 1976/03 | 1976/06 | +3 | Balanced
22 | 158 (large) | 1986/07 | 1986/03 | -4 | Balanced
23 | 121 (medium) | 1996/08 | 1996/08 | 0 | South

The lack of any substantial cycle amplitude dependence on the centroid
positions when time is measured relative to the curve fitted cycle starting
time suggests that the equatorward drift of the sunspot zones follows a
standard path or law. This path is well represented by an exponential function
with

$\bar{\lambda}(t)=28^{\circ}\exp\left[-(t-t_{0})/90\right]$ (4)

where time, $t$, is measured in months. This exponential fit and the data for
the small, medium, and large cycles are plotted as functions of time from the
cycle starting time in the lower panel of Figure 5.

Hemispheric differences in solar activity were first noted by Sporer89 not
long after the discovery of the sunspot cycle itself. Much has been made of
these differences and their possible connection to a variety of sunspot cycle
phenomena. NortonGallagher10 recently revisited these connections and found
little evidence for any of them. Nonetheless we are compelled to examine
possible differences in the sunspot zone locations and equatorward drift
relative to the hemispheric asymmetries. We keep the same starting time for
each hemisphere of each cycle as determined from the curve fitting of the
sunspot numbers but separate the data by the strength of the activity in the
hemisphere. Using the data shown in NortonGallagher10 for the sunspot area
maximum and total sunspot area for each hemisphere in each cycle we identify
cycles in which the northern hemisphere dominates as Cycles 15, 16, 19, and
20, cycles in which the southern hemisphere dominates as Cycles 12, 13, 18,
and 23 with Cycles 14, 17, 21, and 22 having fairly balanced hemispheric
activity. (The relevant sunspot cycle characteristics are listed in Table 1.)
This gives 8 stronger hemispheres, 8 weaker hemispheres, and 8 balanced
hemispheres. The latitude positions of the sunspot zones for the stronger
hemispheres, weaker hemispheres,and balanced hemispheres are shown separately
in Figure 6. We find no significant differences in the sunspot zone latitude
positions associated with hemispheric asymmetry in spite of the fact that for
the unbalanced cycles the same starting time is used for both the strong and
the weak hemisphere.

Figure 6.: The average centroid positions of the sunspot zones for weaker
hemispheres (blue) balanced hemispheres (green) and stronger hemispheres (red)
plotted at 6-month intervals in time from sunspot cycle start. The average
centroid positions for all of the data are shown with $2\sigma$ error bars.
Here too, all three curves fall within the $2\sigma$ errors, criss-crossing
each other along a common, standard trajectory given by the exponential fit in
Equation 4 (dashed line).

##  4 Cycle 23

Cycle 23 had a long, low, extended minimum prior to the start of Cycle 24.
This delayed start of Cycle 24 left behind the lowest smoothed sunspot number
minimum and the largest number of spotless days in nearly a century. The polar
fields during this minimum were the weakest seen in the four cycle record and
the flux of galactic cosmic rays was the highest seen in the six cycle record.

One explanation for both the weak polar fields and the long cycle has been
suggested by flux transport dynamos [Nandy, Muñoz-Jaramillo, & Martens
(2011)]. This model can produce both these characteristics if the meridional
flow was faster than average during the first half of Cycle 23 and slower than
average during the second half. The meridional flow measured by the motions of
magnetic elements in the near surface layers (HathawayRightmire10 and
HathawayRightmire11) exhibited the opposite behavior - slow meridional flow at
the beginning of Cycle 23 and fast meridional flow at the end. Although the
speed of the near surface meridional flow was used to estimate the speed of
the proposed deep meridional return flow in their flux transport dynamo
models, Nandy_etal11 suggest that the variations seen near the surface are
unrelated to variations at the base of the convection zone. However, with
their model the latitudinal drift of the sunspot zones during Cycle 23 should
provide a direct measure of the deep meridional flow and its variations.

Figure 7.: The average centroid positions for Cycle 23 are shown with the
solid line and the exponential fit is shown with the dashed line. The average
centroid positions for all of the data are shown with $2\sigma$ error bars.
Cycle 23 data falls within the $2\sigma$ errors for the full dataset and
follows the standard trajectory. The trajectory (fast then slow) suggested by
Nandy et al. (2011) is shown by the red line. The trajectory (slow then fast)
derived from the observed meridional flow variations (Hathaway & Rightmire
2010, 2011) is shown by the green line.

Figure 7 shows the latitudinal positions of the sunspot zones for Cycle 23
along with those for the full 12 cycle dataset (with $2\sigma$ error bars).
The latitudinal drift of the sunspot zones during Cycle 23 follows within the
$2\sigma$ error range for the average of the last 12 cycles and follows the
standard exponential given by Equation 4. A drift rate that was 30% higher
than average at the start and 30% lower than average at the end of Cycle 23
(the red line in Figure 7) as proposed by Nandy_etal11 is inconsistent with
the data. A drift rate governed by the observed meridional flow variations in
the near surface layers (Hathaway & Rightmire 2010, 2011 - the green line in
Figure 7) is also inconsistent with the data for Cycle 23. This indicates that
the meridional flow is not connected to the latitudinal drift of the sunspot
zones.

##  5 Conclusions

We find that if time is measured relative to a cycle starting time determined
by fitting the monthly sunspot numbers to a parametric curve, then the
latitude positions of the sunspot zones follow a standard path with time. We
find no significant variations from this path associated with sunspot cycle
amplitude or hemispheric asymmetry.

This standard behavior suggests that the equatorward drift of the sunspot
zones is not produced by the Sun’s meridional flow - which is observed (and
theorized) to vary substantially from cycle-to-cycle. This regularity thus
questions the viability of flux transport dynamos as models of the Sun’s
activity cycle. The lack of the variations in drift rate during Cycle 23 in
spite of observed and theorized variations in the meridional flow also argues
against these models.

The earlier kinematic dynamo models of Babcock61 and Leighton69 may be
consistent with the regularity of the sunspot zone drift due to their
dependence on the fairly constant differential rotation profile. However, it
is unclear how the variability of the initial polar fields might influence the
latitudinal drift in these models.

It is clear, however, that this regularity is consistent with dynamo models in
which a Dynamo Wave produces the equatorward drift of the sunspot zones. The
speed of a Dynamo Wave depends on the product of the differential rotation
shear and the kinetic helicity - both of which are not observed or expected to
vary substantially.

### Acknowledgements

The author would like to thank NASA for its support of this research through a
grant from the Heliophysics Causes and Consequences of the Minimum of Solar
Cycle 23/24 Program to NASA Marshall Space Flight Center. He is also indebted
to Lisa Rightmire, Ron Moore, and an anonymous referee whose comments and
suggestions improved both the figures and the manuscript. Most importantly, he
would like to thank the American taxpayers for supporting scientific research
in general and this research in particular.

## References

  * Babcock (1961) Babcock, H. W.: 1961 Astrophys. J. 133 572\.
  * Becker (1954) Becker, U.: 1954 Z. Astrophys. 34 129\.
  * Carrington (1858) Carrington, R. C.: 1858 Mon. Not. Roy. Astron. Soc. 19 1\.
  * Harvey & White (1999) Harvey, K. L. & White, O. R.: 1999 J. Geophys. Res. 104 (A9) 19,759.
  * Hathaway (2010) Hathaway, D. H.: 2010 Living Rev. Solar Phys. 7 1\.
  * Hathaway et al. (2003) Hathaway, D. H., Nandy, D., Wilson, R. M., & Reichmann, R. J.: 2003 Astrophys. J. 589 665\.
  * Hathaway & Rightmire (2010) Hathaway, D. H. & Rightmire, L.: 2010 Science 327 1350\.
  * Hathaway & Rightmire (2011) Hathaway, D. H. & Rightmire, L.: 2011 Astrophys. J. 729 80\.
  * Hathaway, Wilson, & Reichmann (1994) Hathaway, D. H., Wilson, R. M., & Reichmann, R. J.: 1994 Solar Phys. 151 177\.
  * Leighton (1969) Leighton, R. B.: 1969 Astrophys. J. 156 1\.
  * Maunder (1903) Maunder, E. W.: 1903 Observatory 26 329\.
  * Maunder (1904) Maunder, E. W.: 1904 Mon. Not. Roy. Astron. Soc. 64 747\.
  * McKinnon (1987) McKinnon, J. A.: 1987 Sunspot Numbers 1610-1986 (based on The Sunspot-Activity in the Years 1610-1960, by Prof. M. Waldmeier, Copyright 1961 Swiss Federal Observatory, Zurich, Switzerland) UAG Reports UAG-95, National Geophysical Data Center, NOAA, Boulder.
  * Nandy & Choudhuri (2002) Nandy, D. & Choudhuri, A. R.: 2002 Science 296 1671\.
  * Nandy, Muñoz-Jaramillo, & Martens (2011) Nandy, D. & Muñoz-Jaramillo, A., & Martens, P. C. H.: 2011 Nature 471 80\.
  * Norton & Gallagher (2010) Norton, A. A., & , Gallagher, J. C.: 2010 Solar Phys. 261 193\.
  * Press et al. (1986) Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T.: 1986 Numerical Recipes Cambridge University Press, Cambridge, 181pp.
  * Spörer (1880) Spörer, G.: 1880 Publicationen des Astrophysikalischen zu Potsdam 2 No. 5 1.
  * Spörer (1889) Spörer, G.: 1889 Bulletin Astronomique, Serie I 6 60\.
  * Waldmeier (1935) Waldmeier, M.: 1935 Astron. Mitt. Zurich 14 (133) 105.
  * Waldmeier (1955) Waldmeier, M.: 1955 Ergebnisse und Probleme der Sonnenforschung Geest & Portig, Leipzig, 2nd edn.
  * Waldmeier (1961) Waldmeier, M.: 1961 The Sunspot-Activity in the Years 1610-1960 Schulthess Co., Swiss Federal Observatory, Zurich.
  * Yoshimura (1975) Yoshimura, H.: 1975 Astrophys. J. 201 740\.