Griffiths inequality

Short description: Correlation inequality in statistical mechanics

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 $\sigma =\{{\sigma }_j{\}}_{j\in \mathrm{\Lambda }}$ be a configuration of (continuous or discrete) spins on a lattice $\mathrm{\Lambda }$. If $A\subset \mathrm{\Lambda }$ is a list of lattice sites, possibly with duplicates, let ${\sigma }_A=\prod _{j\in A}{\sigma }_j$ be the product of the spins in $A$.

Assign an a-priori measure $d\mu (\sigma )$ 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,

$⟨f⟩=\frac{1}{Z}\int d\mu (\sigma )f(\sigma )e^{-H(\sigma )}$

stands for the ensemble average.

The system is called ferromagnetic if, for any list of sites $A$, $J_A\ge 0$. The system is called invariant under spin flipping if, for any $j$ in $\mathrm{\Lambda }$, the measure $\mu$ is preserved under the sign flipping map $\sigma \to \tau$, where

${\tau }_k=\{\begin{array}{ll}{\sigma }_k,& k\ne j,\\ -{\sigma }_k,& k=j.\end{array}$

Statement of inequalities

First Griffiths inequality

In a ferromagnetic spin system which is invariant under spin flipping,

$⟨{\sigma }_A⟩\ge 0$

for any list of spins A.

Second Griffiths inequality

In a ferromagnetic spin system which is invariant under spin flipping,

$⟨{\sigma }_A{\sigma }_B⟩\ge ⟨{\sigma }_A⟩⟨{\sigma }_B⟩$

for any lists of spins A and B.

The first inequality is a special case of the second one, corresponding to B = $\varnothing$.

Proof

Observe that the partition function is non-negative by definition.

Proof of first inequality: Expand

$e^{-H(\sigma )}=\prod _B\sum _{k\ge 0}\frac{J_B^k{\sigma }_B^k}{k!}=\sum _{\{k_C{\}}_C}\prod _B\frac{J_B^{k_B}{\sigma }_B^{k_B}}{k_B!},$

then

$\begin{array}{rl}Z⟨{\sigma }_A⟩& =\int d\mu (\sigma ){\sigma }_A{e}^{-H(\sigma )}=\sum _{\{k_C{\}}_C}\prod _B\frac{J_B^{k_B}}{k_B!}\int d\mu (\sigma ){\sigma }_A{\sigma }_B^{k_B}\\ & =\sum _{\{k_C{\}}_C}\prod _B\frac{J_B^{k_B}}{k_B!}\int d\mu (\sigma )\prod _{j\in \mathrm{\Lambda }}{\sigma }_j^{n_A(j)+k_B{n}_B(j)},\end{array}$

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, $Z⟨{\sigma }_A⟩\ge 0$, hence also $⟨{\sigma }_A⟩\ge 0$.

Proof of second inequality. For the second Griffiths inequality, double the random variable, i.e. consider a second copy of the spin, ${\sigma }^{\prime }$, with the same distribution of $\sigma$. Then

$⟨{\sigma }_A{\sigma }_B⟩-⟨{\sigma }_A⟩⟨{\sigma }_B⟩=⟨⟨{\sigma }_A({\sigma }_B-\sigma {\text{'}}_B)⟩⟩.$

Introduce the new variables

${\sigma }_j={\tau }_j+{{\tau }_j}^{\prime },\sigma {\text{'}}_j={\tau }_j-{{\tau }_j}^{\prime }.$

The doubled system $⟨⟨\cdot ⟩⟩$ is ferromagnetic in $\tau ,{\tau }^{\prime }$ because $-H(\sigma )-H({\sigma }^{\prime })$ is a polynomial in $\tau ,{\tau }^{\prime }$ with positive coefficients

$\begin{array}{rl}\sum _A{J}_A({\sigma }_A+\sigma {\text{'}}_A)& =\sum _A{J}_A\sum _{X\subset A}[1+(-1{)}^{|X|}]{\tau }_{A\setminus X}\tau {\text{'}}_X\end{array}$

Besides the measure on $\tau ,{\tau }^{\prime }$ is invariant under spin flipping because $d\mu (\sigma )d\mu ({\sigma }^{\prime })$ is. Finally the monomials ${\sigma }_A$, ${\sigma }_B-\sigma {\text{'}}_B$ are polynomials in $\tau ,{\tau }^{\prime }$ with positive coefficients

$\begin{array}{rl}{\sigma }_A& =\sum _{X\subset A}{\tau }_{A\setminus X}\tau {\text{'}}_X,\\ {\sigma }_B-\sigma {\text{'}}_B& =\sum _{X\subset B}[1-(-1{)}^{|X|}]{\tau }_{B\setminus X}\tau {\text{'}}_X.\end{array}$

The first Griffiths inequality applied to $⟨⟨{\sigma }_A({\sigma }_B-\sigma {\text{'}}_B)⟩⟩$ 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

$⟨f{⟩}_h=\int f(x)e^{-h(x)}d\mu (x)/\int e^{-h(x)}d\mu (x).$

Let A be a set of real functions on Γ such that. for every f₁,f₂,...,f_n in A, and for any choice of signs ±,

$\iint d\mu (x)d\mu (y)\prod _{j=1}^n(f_j(x)\pm f_j(y))\ge 0.$

Then, for any f,g,−h in the convex cone generated by A,

$⟨fg{⟩}_h-⟨f{⟩}_h⟨g{⟩}_h\ge 0.$

Proof

Let

$Z_h=\int e^{-h(x)}d\mu (x).$

Then

$\begin{array}{rl}& Z_h^2(⟨fg{⟩}_h-⟨f{⟩}_h⟨g{⟩}_h)\\ & =\iint d\mu (x)d\mu (y)f(x)(g(x)-g(y))e^{-h(x)-h(y)}\\ & =\sum _{k=0}^{\mathrm{\infty }}\iint d\mu (x)d\mu (y)f(x)(g(x)-g(y))\frac{(-h(x)-h(y){)}^k}{k!}.\end{array}$

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

$\frac{\partial }{\partial J_B}⟨{\sigma }_A⟩=⟨{\sigma }_A{\sigma }_B⟩-⟨{\sigma }_A⟩⟨{\sigma }_B⟩\ge 0$

Hence $⟨{\sigma }_A⟩$ 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 $\beta /2$

$⟨{\mathbf{𝐬}}_i\cdot {\mathbf{𝐬}}_j{⟩}_{J,2\beta }\le ⟨{\sigma }_i{\sigma }_j{⟩}_{J,\beta }$

Hence the critical $\beta$ of the XY model cannot be smaller than the double of the critical $\beta$ of the Ising model

${\beta }_c^{XY}\ge 2{\beta }_c^{\mathrm{I}\mathrm{s}};$

in dimension D = 2 and coupling J = 1, this gives

${\beta }_c^{XY}\ge \ln (1+\sqrt{2})\approx 0.88.$

• 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.