# Griffiths inequality

In statistical mechanics, the Griffiths inequality, sometimes also called Griffiths–Kelly–Sherman inequality or GKS inequality, named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems. Informally, it says that in ferromagnetic spin systems, if the 'a-priori distribution' of the spin is invariant under spin flipping, the correlation of any monomial of the spins is non-negative; and the two point correlation of two monomial of the spins is non-negative.
The inequality was proved by Griffiths for Ising ferromagnets with two-body interactions, then generalised by Kelly and Sherman to interactions involving an arbitrary number of spins, and then by Griffiths to systems with arbitrary spins. A more general formulation was given by Ginibre, and is now called the Ginibre inequality.

## Definitions

Let $\textstyle \sigma =\{\sigma _{j}\}_{j\in \Lambda }$ be a configuration of (continuous or discrete) spins on a lattice Λ. If A ⊂ Λ is a list of lattice sites, possibly with duplicates, let $\textstyle \sigma _{A}=\prod _{j\in A}\sigma _{j}$ be the product of the spins in A.
Assign an a-priori measure dμ(σ) on the spins; let H be an energy functional of the form
    $H(\sigma )=-\sum _{A}J_{A}\sigma _{A}~,$
where the sum is over lists of sites A, and let
    $Z=\int d\mu (\sigma )e^{-H(\sigma )}$
be the partition function. As usual,
    $\langle f\rangle ={\frac {1}{Z}}\sum _{\sigma }f(\sigma )e^{-H(\sigma )}$
stands for the ensemble average.
The system is called ferromagnetic if, for any list of sites A, J$_A$ ≥ 0. The system is called invariant under spin flipping if, for any j in Λ, the measure μ is preserved under the sign flipping map σ → τ, where
    $\tau _{k}={\begin{cases}\sigma _{k},&k\neq j,\\-\sigma _{k},&k=j.\end{cases}}$

## Statement of inequalities
### First Griffiths inequality

In a ferromagnetic spin system which is invariant under spin flipping,
    $\langle \sigma _{A}\rangle \geq 0$
for any list of spins A.

### Second Griffiths inequality

In a ferromagnetic spin system which is invariant under spin flipping,
    $\langle \sigma _{A}\sigma _{B}\rangle \geq \langle \sigma _{A}\rangle \langle \sigma _{B}\rangle$
for any lists of spins A and B.
The first inequality is a special case of the second one, corresponding to B = ∅.

## Proof

Observe that the partition function is non-negative by definition.
Proof of first inequality: Expand
    $e^{-H(\sigma )}=\prod _{B}\sum _{k\geq 0}{\frac {J_{B}^{k}\sigma _{B}^{k}}{k!}}=\sum _{\{k_{C}\}_{C}}\prod _{B}{\frac {J_{B}^{k_{B}}\sigma _{B}^{k_{B}}}{k_{B}!}}~,$
then
    ${\begin{aligned}Z\langle \sigma _{A}\rangle &=\int d\mu (\sigma )\sigma _{A}e^{-H(\sigma )}=\sum _{\{k_{C}\}_{C}}\prod _{B}{\frac {J_{B}^{k_{B}}}{k_{B}!}}\int d\mu (\sigma )\sigma _{A}\sigma _{B}^{k_{B}}\\&=\sum _{\{k_{C}\}_{C}}\prod _{B}{\frac {J_{B}^{k_{B}}}{k_{B}!}}\int d\mu (\sigma )\prod _{j\in \Lambda }\sigma _{j}^{n_{A}(j)+k_{B}n_{B}(j)}~,\end{aligned}}$
where n$_A$(j) stands for the number of times that j appears in A. Now, by invariance under spin flipping,
    $\int d\mu (\sigma )\prod _{j}\sigma _{j}^{n(j)}=0$
if at least one n(j) is odd, and the same expression is obviously non-negative for even values of n. Therefore, Z<σ$_A$>≥0, hence also <σ$_A$>≥0.
Proof of second inequality. For the second Griffiths inequality, double the random variable, i.e. consider a second copy of the spin, $\sigma '$, with the same distribution of $\sigma$. Then
    $\langle \sigma _{A}\sigma _{B}\rangle -\langle \sigma _{A}\rangle \langle \sigma _{B}\rangle =\langle \langle \sigma _{A}(\sigma _{B}-\sigma '_{B})\rangle \rangle ~.$
Introduce the new variables
    $\sigma _{j}=\tau _{j}+\tau _{j}'~,\qquad \sigma '_{j}=\tau _{j}-\tau _{j}'~.$
The doubled system $\langle \langle \;\cdot \;\rangle \rangle$ is ferromagnetic in $\tau ,\tau '$ because $-H(\sigma )-H(\sigma ')$ is a polynomial in $\tau ,\tau '$ with positive coefficients
    ${\begin{aligned}\sum _{A}J_{A}(\sigma _{A}+\sigma '_{A})&=\sum _{A}J_{A}\sum _{X\subset A}\left[1+(-1)^{|X|}\right]\tau _{A\setminus X}\tau '_{X}\end{aligned}}$
Besides the measure on $\tau ,\tau '$ is invariant under spin flipping because $d\mu (\sigma )d\mu (\sigma ')$ is. Finally the monomials $\sigma _{A}$, $\sigma _{B}-\sigma '_{B}$ are polynomials in $\tau ,\tau '$ with positive coefficients
    ${\begin{aligned}\sigma _{A}&=\sum _{X\subset A}\tau _{A\setminus X}\tau '_{X}~,\\\sigma _{B}-\sigma '_{B}&=\sum _{X\subset B}\left[1-(-1)^{|X|}\right]\tau _{B\setminus X}\tau '_{X}~.\end{aligned}}$
The first Griffiths inequality applied to $\langle \langle \sigma _{A}(\sigma _{B}-\sigma '_{B})\rangle \rangle$ gives the result.
More details are in and.

## Extension: Ginibre inequality

The Ginibre inequality is an extension, found by Jean Ginibre, of the Griffiths inequality.

### Formulation

Let (Γ, μ) be a probability space. For functions f, h on Γ, denote
    $\langle f\rangle _{h}=\int f(x)e^{-h(x)}\,d\mu (x){\Big /}\int e^{-h(x)}\,d\mu (x).$
Let A be a set of real functions on Γ such that. for every f$_1$,f$_2$,...,f$_n$ in A, and for any choice of signs ±,
    $\iint d\mu (x)\,d\mu (y)\prod _{j=1}^{n}(f_{j}(x)\pm f_{j}(y))\geq 0.$
Then, for any f,g,−h in the convex cone generated by A,
    $\langle fg\rangle _{h}-\langle f\rangle _{h}\langle g\rangle _{h}\geq 0.$

### Proof

Let
    $Z_{h}=\int e^{-h(x)}\,d\mu (x).$
Then
    ${\begin{aligned}&Z_{h}^{2}\left(\langle fg\rangle _{h}-\langle f\rangle _{h}\langle g\rangle _{h}\right)\\&\qquad =\iint d\mu (x)\,d\mu (y)f(x)(g(x)-g(y))e^{-h(x)-h(y)}\\&\qquad =\sum _{k=0}^{\infty }\iint d\mu (x)\,d\mu (y)f(x)(g(x)-g(y)){\frac {(-h(x)-h(y))^{k}}{k!}}.\end{aligned}}$
Now the inequality follows from the assumption and from the identity
    $f(x)={\frac {1}{2}}(f(x)+f(y))+{\frac {1}{2}}(f(x)-f(y)).$

### Examples

* To recover the (second) Griffiths inequality, take Γ = {−1, +1}$^Λ$, where Λ is a lattice, and let μ be a measure on Γ that is invariant under sign flipping. The cone A of polynomials with positive coefficients satisfies the assumptions of the Ginibre inequality.
* (Γ, μ) is a commutative compact group with the Haar measure, A is the cone of real positive definite functions on Γ.
* Γ is a totally ordered set, A is the cone of real positive non-decreasing functions on Γ. This yields Chebyshev's sum inequality. For extension to partially ordered sets, see FKG inequality.

## Applications

* The thermodynamic limit of the correlations of the ferromagnetic Ising model (with non-negative external field h and free boundary conditions) exists.
    This is because increasing the volume is the same as switching on new couplings J$_B$ for a certain subset B. By the second Griffiths inequality
       ${\frac {\partial }{\partial J_{B}}}\langle \sigma _{A}\rangle =\langle \sigma _{A}\sigma _{B}\rangle -\langle \sigma _{A}\rangle \langle \sigma _{B}\rangle \geq 0$
    Hence $\langle \sigma _{A}\rangle$ is monotonically increasing with the volume; then it converges since it is bounded by 1.
* The one-dimensional, ferromagnetic Ising model with interactions $J_{x,y}\sim |x-y|^{-\alpha }$ displays a phase transition if $1<\alpha <2$.
    This property can be shown in a hierarchical approximation, that differs from the full model by the absence of some interactions: arguing as above with the second Griffiths inequality, the results carries over the full model.
* The Ginibre inequality provides the existence of the thermodynamic limit for the free energy and spin correlations for the two-dimensional classical XY model. Besides, through Ginibre inequality, Kunz and Pfister proved the presence of a phase transition for the ferromagnetic XY model with interaction $J_{x,y}\sim |x-y|^{-\alpha }$ if $2<\alpha <4$.
* Aizenman and Simon used the Ginibre inequality to prove that the two point spin correlation of the ferromagnetic classical XY model in dimension $D$, coupling $J>0$ and inverse temperature $\beta$ is dominated by (i.e. has upper bound given by) the two point correlation of the ferromagnetic Ising model in dimension $D$, coupling $J>0$, and inverse temperature $\beta /2$
       $\langle \mathbf {s} _{i}\cdot \mathbf {s} _{j}\rangle _{J,2\beta }\leq \langle \sigma _{i}\sigma _{j}\rangle _{J,\beta }$
    Hence the critical $\beta$ of the XY model cannot be smaller than the double of the critical $\beta$ of the Ising model
       $\beta _{c}^{XY}\geq 2\beta _{c}^{\rm {Is}}~;$
    in dimension D = 2 and coupling J = 1, this gives
       $\beta _{c}^{XY}\geq \ln(1+{\sqrt {2}})\approx 0.88~.$
* There exists a version of the Ginibre inequality for the Coulomb gas that implies the existence of thermodynamic limit of correlations.
* Other applications (phase transitions in spin systems, XY model, XYZ quantum chain) are reviewed in.