# The strong Markov property of Gibbs measures in 2D Ising Model

My background is that of a mathematician. I have a question about the two Dimensional Ising Model. I think the terminology I use is similar to the physical.

I'm trying to understand the following passage in the article on the arXiv about strong Markov property in two-dimensional Ising model.

The strong Markov property of Gibbs measures, stating that $\mu(\cdot\,|\mathcal{F}_{\Gamma^c})(\omega) =\mu_{\Gamma(\omega)}^\omega$ for $\mu$-almost all $\omega$ when $\Gamma$ is any finite random subset of $\mathbb{Z}^2$ which is em determined from outside, in that $\{\Gamma=\Lambda\}\in\mathcal{F}_{\Lambda^c}$ for all finite $\Lambda$, and $\mathcal{F}_{\Gamma^c}$ is the $\sigma$-algebra of all events $A$ outside $\Gamma$, in the sense that $A\cap \{\Gamma=\Lambda\}\in\mathcal{F}_{\Lambda^c}$ for all finite $\Lambda$. (Using the conventions $\mu_{\emptyset}^\omega=\delta_\omega$ and $\mathcal{F}_{\emptyset^c}=\mathcal{F}$ we can in fact allow that $\Gamma$ takes the value $\emptyset$.) For a proof
one simply splits $\Omega$ into the disjoint sets $\{\Gamma=\Lambda\}$ for finite $\Lambda$.

Could someone explain to me clearly an intuition about this property Strong Two-dimensional Ising model and the respective proof?

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It is a direct consequence of the usual (weak) spatial Markov property enjoyed by the Ising model. Namely, if $\Lambda$ is a (deterministic) finite subset of $\mathbb{Z}^d$, and $f$ a local function with support inside $\Lambda$, then the expected value of $f$ in the box $\Lambda$, with a given frozen configuration $\omega$ outside $\Lambda$ only depends on the value taken by $\omega$ along the boundary of $\Lambda$ (assuming nearest-neighbor interactions, as in the linked paper). This follows immediately from the DLR equation.

The strong Markov property is the analogous property when $\Lambda$ is a random finite subset of $\mathbb{Z}^d$. In order for the property to be satisfied in this case, it is necessary that knowing $\Lambda$ only imposes conditions on the spins outside $\Lambda$ (otherwise that would have an effect on the expected value of $f$); this is the reason for asking that $\Lambda$ be determined from outside.

An example would be: "Let $\Lambda$ be the outermost circuit of $+$ spins surrounding the support of $f$ in a region $\Delta$ of $\mathbb{Z}^d$". This imposes constraints outside $\Lambda$ (since, e.g., there cannot be a circuit of $+$ spins surrounding $\Lambda$ inside the region $\Delta$), but not inside.

The proof of the strong version is then a corollary of the (weak) Markov property: just decompose the expectation of $f$ according to the realization of the random set $\Lambda$, and, for each realization, apply the weak Markov property.

PS: Forgive the plug, but I'd like to point out two recent works that provide alternative approaches to the Aizenman-Higuchi theorem, and yield considerably stronger results: arXiv:1003.6034 (Ising model) and arXiv:1205.4659 (Potts model).

EDIT: I've looked at your slightly more detailed question on math.stackexchange, but I still don't see precisely where your difficulty lies. So, let me try to give an explicit example. To keep things easy, let us only consider the nearest-neighbor model, and work in the finite box $\Lambda=\{-n,\ldots,n\}^2$ with $+$ boundary conditions. Let us consider the random variable $\sigma_0$ giving the spin at the center of the box. Given a contour $\gamma$ (I assume you know what this is) surrounding the origin, let $\mathcal{E}_\gamma$ be the event that $\gamma$ is the outermost contour surrounding the origin, and $\mathcal{E}_\varnothing$ be the event that no such contour exist. Then the expectation of $\sigma_0$ can be decomposed according to $\gamma$: $$\mu^+_\Lambda(\sigma_0) = \sum_\gamma \mu^+_\Lambda(\mathcal{E}_\gamma) \mu^+_\Lambda(\sigma_0 \,|\, \mathcal{E}_\gamma),$$ where the sum is over all contours $\gamma$ in $\Lambda_n$ surrounding the origin, including the degenerate case $\gamma=\emptyset$.

The point, now, is that, when $\gamma\neq\emptyset$, $$\mu^+_\Lambda(\sigma_0 \,|\, \mathcal{E}_\gamma) = \mu^-_{\rm int(\gamma)}(\sigma_0),$$ since, conditionnally on $\mathcal{E}_\gamma$, there is a path of $-$ spins along $\gamma$ (on the "inner" side of $\gamma$), which decouples the configuration inside $\gamma$ from the configuration outside. Note, and this is crucial, that the fact that we chose the outermost contour implies that the configuration inside $\rm int(\gamma)$ is unconstrained (while the configuration outside $\gamma$ is very much constrained, since it must not destroy the fact that $\gamma$ was outermost).

The last identity is the strong Markov property for the finite-volume nearest-neighbor Ising model. The property in your question is the analogous one for the infinite volume Gibbs measure.

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Thanks for answering my pedantic question . I will study your answer. But as I know that You is a reference in the subject I will now point out and accept your answer. Thank you. – MathOverview Dec 11 '12 at 11:52
@Elias: of course, if you need additonal informations, don't hesitate to ask (it is never clear to what level of details one needs to go, and it also very much depends on where the confusion lies). You might also do that by email, my adress can be found on my homepage. – Yvan Velenik Dec 11 '12 at 18:15
if you want to receive a bonus of 100 points for your answer in math.stackexchange.com can replicate it in math.stackexchange.com/questions/254018/…‌​‌​ov-propert-in-2d-ising-model . It will be a pleasure bonific it. – MathOverview Dec 12 '12 at 11:25
Thank you for your attention Velenik. Thank you. – MathOverview Dec 12 '12 at 11:28
@Elias: I've added some additional explanations in a simpler settings. If this does not clarify the issue, then just tell me... – Yvan Velenik Dec 12 '12 at 14:04