Gibbs measure

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The definition of a Gibbs random field on a lattice requires some terminology:

The lattice: A countable set $\mathbb{L}$ .
The single-spin space: A probability space $(S,\mathcal{S},\lambda)$ .
The configuration space: $(\Omega, \mathcal{F})$ , where $\Omega = S^{\mathbb{L}}$ and $\mathcal{F} = \mathcal{S}^{\mathbb{L}}$ .
Given a configuration $\omega \in \Omega$ and a subset $\Lambda \subset \mathbb{L}$ , the restriction of $ω$ to $Λ$ is $\omega_\Lambda = (\omega(t))_{t\in\Lambda}$ . If $\Lambda_1\cap\Lambda_2=\emptyset$ and $\Lambda_1\cup\Lambda_2=\mathbb{L}$ , then the configuration $\omega_{\Lambda_1}\omega_{\Lambda_2}$ is the configuration whose restrictions to $Λ 1$ and $Λ 2$ are $\omega_{\Lambda_1}$ and $\omega_{\Lambda_2}$ , respectively.
The set $\mathcal{L}$ of all finite subsets of $\mathbb{L}$ .
For each subset $\Lambda\subset\mathbb{L}$ , $\mathcal{F}_\Lambda$ is the $σ$ -algebra generated by the family of functions $(\sigma(t))_{t\in\Lambda}$ , where $σ(t)(ω) = ω(t)$ .
The potential: A family of functions such that
1. For each $A\in\mathcal{L}$ , $Φ A$ is $\mathcal{F}_A$ -measurable.
2. For all $\Lambda\in\mathcal{L}$ and $\omega\in\Omega$ , the series $H_\Lambda^\Phi(\omega) = \sum_{A\in\mathcal{L}, A\cap\Lambda\neq\emptyset} \Phi_A(\omega)$ exists.
The Hamiltonian in $\Lambda\in\mathcal{L}$ with boundary conditions $\bar\omega$ , for the potential $Φ$ , is defined by

$H_\Lambda^\Phi(\omega | \bar\omega) = H_\Lambda^\Phi(\omega_\Lambda\bar\omega_{\Lambda^c})$ ,

where $\Lambda^c = \mathbb{L}\setminus\Lambda$ .

The partition function in $\Lambda\in\mathcal{L}$ with boundary conditions $\bar\omega$ and inverse temperature $\beta\in\mathbb{R}_+$ (for the potential $Φ$ and $λ$ ) is defined by

$Z_\Lambda^\Phi(\bar\omega) = \int \lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega | \bar\omega))$ .

A potential

Φ

λ

-admissible if $Z_\Lambda^\Phi(\bar\omega)$ is finite for all $\Lambda\in\mathcal{L}$ , $\bar\omega\in\Omega$ and

β > 0

A probability measure $μ$ on $(\Omega,\mathcal{F})$ is a Gibbs measure for a $λ$ -admissible potential $Φ$ if it satisfies the Dobrushin-Lanford-Ruelle (DLR) equations

$\int \mu(\mathrm{d}\bar\omega)Z_\Lambda^\Phi(\bar\omega)^{-1} \int\lambda^\Lambda(\mathrm{d}\omega) \exp(-\beta H_\Lambda^\Phi(\omega | \bar\omega)) 1_A(\omega_\Lambda\bar\omega_{\Lambda^c}) = \mu(A)$ ,

for all $A\in\mathcal{F}$ and $\Lambda\in\mathcal{L}$ .

[edit] An example

To help understand the above definitions, here are the corresponding quantities in the important example of the Ising model with nearest-neighbour interactions (coupling constant $J$ ) and a magnetic field ( $h$ ), on $\mathbb{Z}^d$ :

The lattice is simply $\mathbb{L} = \mathbb{Z}^d$ .
The single-spin space is $S = { - 1,1}$ .
The potential is given by

$\Phi_A(\omega) = \begin{cases} -J\,\omega(t_1)\omega(t_2) & \mathrm{if\ } A=\{t_1,t_2\} \mathrm{\ with\ } \|t_2-t_1\|_1 = 1 \\ -h\,\omega(t) & \mathrm{if\ } A=\{t\}\\ 0 & \mathrm{otherwise} \end{cases}$