Dependencies

Let \(\mu \in \mathfrak {M}(X,\mathcal X)\). Then, \(\pi \in \text{Ker}_{\mathcal B,\mathcal X}\) is a conditional expectation kernel for \(\mu \) if \(\mu (A\mid \mathcal B)=\pi (A\mid \cdot )\) \(\mu \)-a.e.

Let \((E,\mathcal{E})\) be a measurable space, and let \(S\) be a set. Then,

defines the cylinder events in \(\Delta \) (for each \(\Delta \in \mathcal{P}(S)\)), where each \(\text{proj}_\delta \) is the coordinate projection at coordinate \(\delta \).

Given a specification \(\gamma \), a Gibbs measures specified by \(\gamma \) is a measure \(\nu \in \mathfrak {M}(E^S, \mathcal{E}^S)\) such that \(\gamma _\Lambda (A|\cdot )\) is a conditional expectation kernel for \(\nu \) for all \(A \in \mathcal E^S\) and \(\Lambda \in \operatorname{Finset}(S)\).

A specification \(\gamma \) is independent iff

The Independent Specification with Single Spin Distribution \(\nu \) is

defines the Independent Specification with Single Spin Distribution with \(\nu \) (for each \(\nu \in \mathfrak {P}(E, \mathcal{E})\)), where \(\nu ^\Lambda \) is the usual product measure.

\(\operatorname{ISSSD}(\nu )\) is a specification.

Let \(E\) and \(S\) be sets. Let \(\Delta \in \mathcal{P}(S)\), and let \(\omega \in E^S\). We define

to be the juxtaposition of \(\zeta \) and \(\omega \) (for each \(\zeta \in E^\Delta \)).

Let \((X,\mathcal X)\) and \((Y,\mathcal{Y})\) be measurable spaces. Then,

defines the set of kernels from \(\mathcal{Y}\) to \(\mathcal X\), where \(\mathfrak {M}(X,\mathcal X)\) is the space of measures on \(X\).

The conditional expectation of a \(\mathcal X\)-measurable function \(f : X \to [0, \infty ]\) is

Let \((X,\mathcal X)\) and \((Y,\mathcal{Y})\) be measurable spaces. We say that \(\pi \in \text{Ker}_{\mathcal{Y},\mathcal X}\) is a Markov kernel iff \(\pi (X\mid \cdot )=1\).

A specification \(\gamma \) is a Markov specification iff \(\gamma _\Lambda \) is a probability kernel for every \(\Lambda \in \operatorname{Finset}(S)\).

A modifier of \(\gamma \) is a family

such that the corresponding family of kernels \(\rho \gamma \) is a specification.

A family of measurable functions \(h_\Lambda : (S \to E) \to [0, \infty [\) is a premodifier if

for all \(\Lambda _1 \subseteq \Lambda _2\) and all \(\zeta , \eta : S \to E\) such that \(\zeta _{\Lambda _1^c} = \eta _{\Lambda _1^c}\).

Let \(I\) be a set. Suppose for each \(i\in I\) that \((\Omega _i,\mathcal B_i,P_i)\) is a probability space. Then, \(P:=\bigotimes _{i\in I}P_i\) is a well-defined product probability measure on \(\prod _{i\in I}\Omega _i\).

\(\pi \) is proper iff \(\pi (A\cap B\mid x)=\pi (A\mid x)\cdot \mathbf{1}_B(x)\) for all \(A\in \mathcal X\), \(B\in \mathcal B\) and \(x\in X\).

A specification \(\gamma \) is proper iff the kernel \(\gamma _\Lambda \) is proper for every \(\Lambda \in \operatorname{Finset}(S)\).

A specification is a family of kernels \(\gamma : \operatorname{Finset}S \to \operatorname{Ker}_{\mathcal{F}_{S\setminus \Lambda }, \mathcal{E}^S}\) which is consistent, in the sense that

Let \(\mu \in \mathfrak {M}(X,\mathcal X)\) be a finite measure and let \(\pi \in \text{Ker}_{\mathcal B,\mathcal X}\) be a proper kernel. Then,

If \(\pi \in \text{Ker}_{\mathcal B, \mathcal X}\) is a conditional expectation kernel for \(\mu \), then

for all bounded \(\mathcal X\)-measurable functions \(f : X \to \mathbb R\).

If \(\pi \in \text{Ker}_{\mathcal B, \mathcal X}\) is a conditional expectation kernel for \(\mu \), then

for all \(\mathcal X\)-measurable functions \(f : X \to [0, \infty ]\).

If \(E\) is countable, \(\nu \) is the counting measure and \(\gamma \) is any specification, then

is a modifier from \(\operatorname{ISSSD}(\nu )\) to \(\gamma \).

Let \(\gamma \) be a proper specification with parameter set \(S\) and state space \((E, \mathcal{E})\), and let \(\nu \in \mathfrak {P}(E^S, \mathcal{E}^S)\). TFAE:

1. \(\nu \in \mathcal{G}(\gamma )\).

2. \(\gamma _\Lambda \circ _m\nu = \nu \) for all \(\Lambda \in \operatorname{Finset}(S)\).

3. \(\gamma _\Lambda \circ _m\nu = \nu \) frequently as \(\Lambda \to S\).

The product measure \(\nu ^S\) is a Gibbs measure specified by \(\operatorname{ISSSD}(\nu )\).

There is at most one Gibbs measure specified by \(\operatorname{ISSSD}(\nu )\).

\(\operatorname{ISSSD}(\nu )\) is independent.

\(\operatorname{ISSSD}(\nu )\) is a proper specification.

If \(f : X \to [0, \infty ]\) is a \(\mathcal X\)-measurable function, then \(\mu [f | \mathcal B]\) is the \(\mu \)-ae unique \(\mathcal B\)-measurable function \(X \to [0, \infty ]\) such that

for all \(B \in \mathcal B\).

Modifying a specification \(\gamma \) by \(\rho _1\) then \(\rho _2\) is the same as modifying it by their product.

If \(\rho \) is a modifier of \(\operatorname{ISSSD}(\nu ^S)\), then it is a premodifier if any of the following conditions hold:

1. \(E\) is countable and \(\nu \) is equivalent to the counting measure.

2. \(E\) is a second countable Borel space.

3. \(\nu \) is everywhere dense.

4. For all \(\Lambda _1 \subseteq \Lambda _2\) and all \(\eta : S \to E\), \(\zeta \mapsto \rho _{\Lambda _1}(\zeta \eta _{\Lambda _1^c})\) is continuous on \(E^{\Lambda _1}\).

If \(\gamma \) is a specification and \(\rho \) a modifier of \(\gamma \), then \(\rho \gamma \) is a proper specification.

If \(h\) is a premodifier and \(\nu \) is such that \(0 {\lt} \nu _\Lambda h_\Lambda {\lt} \infty \) for all \(\Lambda \), then

is a modifier of \(\operatorname{ISSSD}(\nu )\).

If \(\pi \) is a proper Markov kernel, then

for all \(x_0 \in X\) and functions \(f, g : X \to \mathbb R\) such that \(f\) is bounded \(\mathcal X\)-measurable and \(g\) is bounded \(\mathcal B\)-measurable.

If \(\pi \) is proper, then

for all \(x_0 \in X\) and functions \(f, g : X \to [0, \infty ]\) such that \(f\) is \(\mathcal X\)-measurable, \(g\) is \(\mathcal B\)-measurable.

If \(\rho \) is a family of measurable densities and \(\gamma \) is a proper specification, then TFAE

1. \(\rho \) is a modifier of \(\gamma \)

2. For all \(\Lambda _1, \Lambda _2\) with \(\Lambda _1 \subseteq \Lambda _2\) and all \(\eta : S \to E\), we have

   \[ \rho _{\Lambda _2} = \rho _{\Lambda _1}\cdot (\gamma _{\Lambda _1} \rho _{\Lambda _2}) \quad \gamma _{\Lambda _2}(\cdot |\eta )\text{-a.e.} \]

If \(\rho \) is a family of measurable densities and \(\gamma \) is a proper independent specification, then TFAE

1. \(\rho \) is a modifier of \(\gamma \)

2. For all \(\Lambda _1, \Lambda _2\) with \(\Lambda _1 \subseteq \Lambda _2\), \(\eta : S \to E\) and \(\gamma _{\Lambda _2 \setminus \Lambda _1}(\cdot |\alpha )\)-almost all \(\eta _2 : S \to E\), we have

   \[ \rho _{\Lambda _2}(\zeta _1)\rho _{\Lambda _1}(\zeta _2) = \rho _{\Lambda _2}(\zeta _2) \rho _{\Lambda _1}(\zeta _1) \]

   for \(\gamma _{\Lambda _1}(\cdot |\eta _2) \times \gamma _{\Lambda _2}(\cdot |\eta _2)\)-almost all \((\zeta _1, \zeta _2)\).