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 be a configuration of (continuous or discrete) spins on a lattice . If is a list of lattice sites, possibly with duplicates, let be the product of the spins in .

Assign an a-priori measure on the spins; let be an energy functional of the form

where the sum is over lists of sites , and let

be the partition function. As usual,

stands for the ensemble average.

The system is called ferromagnetic if, for any list of sites , . The system is called invariant under spin flipping if, for any in , the measure is preserved under the sign flipping map , where

Statement of inequalities

First Griffiths inequality

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

for any list of spins A.

Second Griffiths inequality

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

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

then

where n_A(j) stands for the number of times that j appears in A. Now, by invariance under spin flipping,

if at least one n(j) is odd, and the same expression is obviously non-negative for even values of n. Therefore, , hence also .

Proof of second inequality. For the second Griffiths inequality, double the random variable, i.e. consider a second copy of the spin, , with the same distribution of . Then

Introduce the new variables

The doubled system is ferromagnetic in because is a polynomial in with positive coefficients

Besides the measure on is invariant under spin flipping because is. Finally the monomials , are polynomials in with positive coefficients

The first Griffiths inequality applied to 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, ÃŽÂ¼) be a probability space. For functions f, h on Γ, denote

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 ±,

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

Proof

Let

Then

Now the inequality follows from the assumption and from the identity

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, ÃŽÂ¼) 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

:

Hence is monotonically increasing with the volume; then it converges since it is bounded by 1.

• The one-dimensional, ferromagnetic Ising model with interactions displays a phase transition if .

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 if .
• Aizenman and Simon used the Ginibre inequality to prove that the two point spin correlation of the ferromagnetic classical XY model in dimension , coupling and inverse temperature is dominated by (i.e. has upper bound given by) the two point correlation of the ferromagnetic Ising model in dimension , coupling , and inverse temperature

:

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

:

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

:

• 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.

References