1 Specifications of random fields
1.1 Preliminaries
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 \((E,\mathcal{E})\) 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 \(\delta \).
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 \(X\).
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 \(\pi (X\mid \cdot )=1\).
Let \((X, \mathcal X)\) be a measurable space, let \(\mathcal B\) be a sub \(\sigma \)-algebra of \(\mathcal X\). Let \(\pi \in \text{Ker}_{\mathcal B, \mathcal X}\).
\(\pi \) 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 \(x\in X\).
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.
Standard machine.
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.
Standard machine.
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.
If \(\pi \in \text{Ker}_{\mathcal B, \mathcal X}\) is a conditional expectation kernel for \(\mu \), then
for all \(\mathcal X\)-measurable functions \(f : X \to [0, \infty ]\).
Standard machine.
If \(\pi \in \text{Ker}_{\mathcal B, \mathcal X}\) is a conditional expectation kernel for \(\mu \), then
for all bounded \(\mathcal X\)-measurable functions \(f : X \to \mathbb R\).
Standard machine.
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,
By the characterisation of conditional expectation,
By properness of \(\pi \),
Hence
1.2 Prescribing conditional probabilities
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
All specifications will be with parameter set \(S\) and state space \((E, \mathcal{E})\) in this chapter.
A specification \(\gamma \) is independent iff
A specification \(\gamma \) is a Markov specification iff \(\gamma _\Lambda \) is a probability kernel for every \(\Lambda \in \operatorname{Finset}(S)\).
A specification \(\gamma \) is proper iff the kernel \(\gamma _\Lambda \) is proper for every \(\Lambda \in \operatorname{Finset}(S)\).
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 \(\Lambda \in \operatorname{Finset}(S)\).
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:
\(\nu \in \mathcal{G}(\gamma )\).
\(\gamma _\Lambda \circ _m\nu = \nu \) for all \(\Lambda \in \operatorname{Finset}(S)\).
\(\gamma _\Lambda \circ _m\nu = \nu \) frequently as \(\Lambda \to S\).
1 is equivalent to 2 by Lemma 1.9. 2 trivially implies 3. Now, 3 implies 2 because for each \(\Lambda \) there exists some \(\Lambda ' \supseteq \Lambda \) such that \(\gamma _{\Lambda '}\circ _k\nu = \nu \). Then
1.3 \(\lambda \)-specifications
Let \(S\) be a set, \((E, \mathcal{E})\) be a measurable space and \(\nu \) a measure on \(E\).
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 \(\prod _{i\in I}\Omega _i\).
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 independent.
Immediate.
\(\operatorname{ISSSD}(\nu )\) is a specification.
\(\operatorname{ISSSD}(\nu )\) is a proper specification.
We already know it’s a specification. Properness is immediate.
There is at most one Gibbs measure specified by \(\operatorname{ISSSD}(\nu )\).
See book.
The product measure \(\nu ^S\) is a Gibbs measure specified by \(\operatorname{ISSSD}(\nu )\).
Immediate.
A modifier of \(\gamma \) is a family
such that the corresponding family of kernels \(\rho \gamma \) is a specification.
Modifying a specification \(\gamma \) by \(\rho _1\) then \(\rho _2\) is the same as modifying it by their product.
TODO
If \(\gamma \) is a specification and \(\rho \) a modifier of \(\gamma \), then \(\rho \gamma \) is a proper specification.
For all \(\Lambda \in \operatorname{Finset}(S)\), \(A \in \mathcal E^S\), \(B \in \mathcal{F}_{S\setminus \Lambda }\), \(\eta : S \to E\), we want to prove
Expanding out, this is equivalent to
which is true by Lemma 1.6 with \(f = 1_A\rho _\Lambda \), \(g = 1_B\).
If \(E\) is countable, \(\nu \) is the counting measure and \(\gamma \) is any specification, then
is a modifier from \(\operatorname{ISSSD}(\nu )\) to \(\gamma \).
For all \(\Lambda \in \operatorname{Finset}(S)\), \(A\) measurable, \(\eta : S \to E\), we have
If \(\rho \) is a family of measurable densities and \(\gamma \) is a proper specification, then TFAE
\(\rho \) is a modifier of \(\gamma \)
For all \(\Lambda _1, \Lambda _2\) with \(\Lambda _1 \subseteq \Lambda _2\) and all \(\eta : S \to E\), we have
\[ \rho _{\Lambda _2} = \rho _{\Lambda _1}\cdot (\gamma _{\Lambda _1} \rho _{\Lambda _2}) \quad \gamma _{\Lambda _2}(\cdot |\eta )\text{-a.e.} \]
(\(\implies \)) \(\rho _{\Lambda _2} = \rho _{\Lambda _1}\cdot (\gamma _{\Lambda _1} \rho _{\Lambda _2}) \quad \gamma _{\Lambda _2}(\cdot |\eta )\text{-a.e.}\)
\(\implies \rho _{\Lambda _2}\gamma _{\Lambda _2} = \)
If \(\rho \) is a family of measurable densities and \(\gamma \) is a proper independent specification, then TFAE
\(\rho \) is a modifier of \(\gamma \)
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 \((\zeta _1, \zeta _2)\).
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 \(\zeta _{\Lambda _1^c} = \eta _{\Lambda _1^c}\).
If \(\rho \) is a modifier of \(\operatorname{ISSSD}(\nu ^S)\), then it is a premodifier if any of the following conditions hold:
\(E\) is countable and \(\nu \) is equivalent to the counting measure.
\(E\) is a second countable Borel space.
\(\nu \) is everywhere dense.
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 \(E^{\Lambda _1}\).
Use Proposition ??.
Omitted.
Omitted.
Omitted.
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 \(\operatorname{ISSSD}(\nu )\).
TODO