A new correlation inequality for Ising models with external fields

Abstract

We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field, where a subset of vertices is designated as the boundary. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically 0. One corollary is that spin–spin correlations are maximised when the external field vanishes and the boundary condition is free, which proves a conjecture of Shlosman. In particular, the random field Ising model on ${\mathbb {Z}}^d$, $d\geqslant 3$, exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model in $d\geqslant 3$ satisfies the conjectured strong spatial mixing property in the entire high temperature regime.

Appendix: Some counterexamples

In this section, we give the counterexamples mentioned in Remarks 1.2 and 1.4.

Counterexample for Remark 1.2. Consider $V=\{-2,-1,0,1,2\}$ with edges between neighbouring integers and $J_e=1$ for every edge e. Let $g_{-2}=g_{-1}=-2$ and $g_0=0$. It is possible to choose $g_1<2<g_2$ such that the effective field on $\sigma _0$ induced by spins to its left (cf. (2.4) below) exactly cancels out the field induced by spins to its right. If we choose $h_i= 1_{\{i=0\}}$, then $\langle \sigma _0\rangle _{g+h}-\langle \sigma _0\rangle _{g-h}=\langle \sigma _0\rangle _{h}-\langle \sigma _0\rangle _{-h}$ already achieves the maximum. Changing g to $\lambda g$, say for $\lambda \in (0,1)$, in general breaks the balance between the effective fields induced on $\sigma _0$ by spins to its left and right, which leads to strictly smaller values of $\langle \sigma _0\rangle _{\lambda g+h}-\langle \sigma _0\rangle _{\lambda g-h}$ [see (2.2)]. $\square$

Counterexample for Remark 1.4. Consider a tree with root u and three leaves v, a, b. Assume that $J_{ua}\in (0,1)$ and $J_{uv}=J_{ub}=1$. Consider ${\widetilde{g}}$ with ${\widetilde{g}}_u= {\widetilde{g}}_v=0$ and ${\widetilde{g}}_b=1$. Then we can find ${\widetilde{g}}_a<-1$ such that the effective field on $\sigma _u$ induced by $\sigma _a$ [cf. (2.4)] exactly cancels out the field induced by $\sigma _b$, which implies that equality holds in (1.4) with $g={\widetilde{g}}$. If we replace $g=\lambda {\widetilde{g}}$, then because the effective fields induced by $\sigma _a$ and $\sigma _b$ on $\sigma _u$ are distinct non-linear functions of $\lambda$, we can find $\lambda _0\in (0,1)$ such that $\sigma _a$ and $\sigma _b$ together induce a non-zero effective field on $\sigma _u$. Under the external field $\lambda _0 {\widetilde{g}}$, inequality in (1.4) can be seen to be strict. This implies that the l.h.s. of (1.4) with $g=\lambda {\widetilde{g}}$ is not monotonically decreasing in $\lambda \geqslant 0$. We can also construct an example with $J\equiv 1$ by inserting a vertex ${\widetilde{a}}$ between u and a. This has the same effect as having an effective coupling $J_{ua}\in (0,1)$ between u and a with $e^{2J_{ua}}= \cosh 2,$² which is just the example we already constructed. $\square$