Finite-spin Dobrushin influence matrix

Let $d\ge 1$, let $S$ be a finite nonempty set, and put $X=S^{{\mathbb{Z}}^d}$ with the $\sigma$-algebra generated by finite-coordinate cylinders. A specification on $X$ is a family $\gamma =({\gamma }_{\mathrm{\Lambda }})$ of probability kernels indexed by finite $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$. It is proper when each kernel leaves every exterior-measurable event fixed, and consistent when ${\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}={\gamma }_{\mathrm{\Lambda }}$ for all finite $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$. For $i\in {\mathbb{Z}}^d$, write ${\gamma }_i(\cdot \mid \eta )$ for its one-site conditional probability vector on $S$, where $\eta \in S^{{\mathbb{Z}}^d\setminus \{i\}}$. For probability vectors $p,q$ on $S$, define their total-variation distance by $\parallel p-q{\parallel }_{\mathrm{T}\mathrm{V}}:=\frac{1}{2}{\sum }_{s\in S}\mathrm{\mid }p(s)-q(s)\mathrm{\mid }$. The influence coefficient of site $j$ on the conditional law at site $i$ is

$c_{ij}^{\gamma }:=\{\begin{array}{ll}0,& i=j,\\ {\displaystyle \underset{\begin{array}{c}\eta ,{\eta }^{\mathrm{\prime }}\in S^{{\mathbb{Z}}^d\setminus \{i\}}\\ {\eta }_k={\eta }_k^{\mathrm{\prime }}\text{ for every }k\mathrm{\notin }\{i,j\}\end{array}}{\sup }\parallel {\gamma }_i(\cdot \mid \eta )-{\gamma }_i(\cdot \mid {\eta }^{\mathrm{\prime }}){\parallel }_{\mathrm{T}\mathrm{V}},}& i\mathrm{\ne }j\mathrm{.}\end{array}$

The matrix $C^{\gamma }=(c_{ij}^{\gamma }{)}_{i,j\in {\mathbb{Z}}^d}$ is the Dobrushin influence matrix of $\gamma$. Its coefficients lie in $[0,1]$; a supremum need not be attained. The Dobrushin row constant is $c_{\gamma }:={\sup }_i{\sum }_j{c}_{ij}^{\gamma }$, with value $+\mathrm{\infty }$ allowed. A common uniform finite influence radius $R$ means $c_{ij}^{\gamma }=0$ whenever $\parallel i-j{\parallel }_{\mathrm{\infty }}>R$.

Lattice Gibbs kernels and DLR states

The arrow $j\to i$ below means that changing spin $j$ can change the conditional law at $i$; its weight is $C_{ij}$. A length-$n$ path from $j$ to $i$ contributes the product of its edge weights to $(C^n{)}_{ij}$. The picture fixes this index convention only; proofs use the matrix definitions above.