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,


 * $$ \langle \cdot \rangle = \frac{1}{Z} \sum_\sigma \cdot(\sigma) e^{-H(\sigma)} $$

stands for the ensemble average.

The system is called ferromagnetic if, for any list of sites A, JA ≥ 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} $$

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{align}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{align}$$

where nA(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<σA>≥0, hence also <σA>≥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{align}

\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{align}$$

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{align}

\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{align}$$

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 f1,f2,...,fn 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,&minus;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{align}

&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{align} $$

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 = {&minus;1, +1}Λ, where Λ is a lattice, and let &mu; 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 JB 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}

\le \langle \sigma_i\sigma_j\rangle_{J,\beta}$$
 * Hence the critical $$\beta$$ of the XY model cannot be smaller than the double of the critical temperature of the Ising model
 * $$ \beta_c^{XY}\ge 2\beta_c^{\rm Is}~;$$
 * in dimension D = 2 and coupling J = 1, this gives
 * $$ \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.