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 spins on a lattice Λ. If A ⊂ Λ is a list of lattice sites, possibly with duplicates, let be the product of the spins in A.
Assign an a-priori measure dμ on the spins;
let H be an energy functional of the form
where the sum is over lists of sites A, and let
be the partition function. As usual,
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

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 nA stands for the number of times that j appears in A. Now, by invariance under spin flipping,
if at least one n 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, , 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 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,...,fn 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 Griffiths inequality, take Γ = 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.
  • 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 exists.
  • The one-dimensional, ferromagnetic Ising model with interactions displays a phase transition if.
  • 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 the two point correlation of the ferromagnetic Ising model in dimension, coupling, and inverse temperature

  • There exists a version of the Ginibre inequality for the Coulomb gas that implies the existence of thermodynamic limit of correlations.
  • Other applications are reviewed in.