Griffiths Inequalities

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 $\textstyle \sigma=\_$ 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 $\textstyle \sigma_A = \prod_ \sigma_j$ 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

:$H(\sigma)=-\sum_ J_A \sigma_A ~,$

where the sum is over lists of sites ''A'', and let

:$Z=\int d\mu(\sigma) e^$

be the partition function. As usual,

:$\langle \cdot \rangle = \frac \sum_\sigma \cdot(\sigma) e^$

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

:$\tau_k = \begin \sigma_k, &k\neq j, \\ - \sigma_k, &k = j. \end$

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

:$e^ = \prod_ \sum_ \frac = \sum_ \prod_B \frac~,$

then

:$\beginZ \langle \sigma_A \rangle &= \int d\mu(\sigma) \sigma_A e^ = \sum_ \prod_B \frac \int d\mu(\sigma) \sigma_A \sigma_B^ \\ &= \sum_ \prod_B \frac \int d\mu(\sigma) \prod_ \sigma_j^~,\end$

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

:$\int d\mu(\sigma) \prod_j \sigma_j^ = 0$

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
\sum_A J_A (\sigma_A+\sigma'_A) &= \sum_A J_A\sum_
\left +(-1)^\right\tau_ \tau'_X
\end

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
\sigma_A &= \sum_ \tau_ \tau'_~, \\
\sigma_B-\sigma'_B &= \sum_
\left -(-1)^\right\tau_ \tau'_X~.
\end

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

: $\langle f \rangle_h = \int f(x) e^ \, d\mu(x) \Big/ \int e^ \, d\mu(x).$

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 ±,

:$\iint d\mu(x) \, d\mu(y) \prod_^n (f_j(x) \pm f_j(y)) \geq 0.$

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

: $Z_h = \int e^ \, d\mu(x).$

Then

: $\begin &Z_h^2 \left( \langle fg\rangle_h - \langle f \rangle_h \langle g \rangle_h \right)\\ &\qquad= \iint d\mu(x) \, d\mu(y) f(x) (g(x) - g(y)) e^ \\ &\qquad= \sum_^\infty \iint d\mu(x) \, d\mu(y) f(x) (g(x) - g(y)) \frac. \end$

Now the inequality follows from the assumption and from the identity
:$f(x) = \frac (f(x)+f(y)) + \frac (f(x)-f(y)).$

Examples

* To recover the (second) 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.
* (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
::$\frac\langle \sigma_A\rangle= \langle \sigma_A\sigma_B\rangle- \langle \sigma_A\rangle \langle \sigma_B\rangle\geq 0$
: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 $J_\sim , x-y, ^$ 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 $J_\sim , x-y, ^$ 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 $\beta/2$
::$\langle \mathbf_i\cdot \mathbf_j\rangle_ \le \langle \sigma_i\sigma_j\rangle_$
:Hence the critical $\beta$ of the XY model cannot be smaller than the double of the critical temperature of the Ising model
::$\beta_c^\ge 2\beta_c^~;$
:in dimension ''D'' = 2 and coupling ''J'' = 1, this gives
::$\beta_c^ \ge \ln(1 + \sqrt) \approx 0.88~.$

* 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.

References

{{Reflist

Inequalities
Statistical mechanics