Quantitative DLR uniqueness, concentration and spatial response — Examples

Assume Countable Choice: every countable family of nonempty sets has a choice function. Let $d\ge 1$, let $X=\{-1,+1{\}}^{{\mathbb{Z}}^d}$ with its cylinder-generated Borel $\sigma$-algebra, fix $\beta >0$ and $J\in \mathbb{R}$, and let $h=(h_i{)}_{i\in {\mathbb{Z}}^d}\in {\mathbb{R}}^{{\mathbb{Z}}^d}$ be a site field. For $i,j\in {\mathbb{Z}}^d$, put $J_{ij}=J$ if $\parallel i-j{\parallel }_1=1$ and $J_{ij}=0$ otherwise. For a finite $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$ and exterior configuration $\eta \in \{-1,+1{\}}^{{\mathrm{\Lambda }}^c}$, write ${\widehat{\sigma }}_i={\sigma }_i$ for $i\in \mathrm{\Lambda }$ and ${\widehat{\sigma }}_i={\eta }_i$ otherwise, and define

$H_{\mathrm{\Lambda }}^h({\sigma }_{\mathrm{\Lambda }}\mid \eta ):=-{\sum }_{\begin{array}{c}\{i,j\}\subset {\mathbb{Z}}^d:\parallel i-j{\parallel }_1=1,\{i,j\}\cap \mathrm{\Lambda }\mathrm{\ne }\mathrm{\varnothing }\end{array}}J{\widehat{\sigma }}_i{\widehat{\sigma }}_j-{\sum }_{i\in \mathrm{\Lambda }}{h}_i{\sigma }_i\mathrm{.}$

The pair sum is over unordered pairs and is finite because each site has $2d$ nearest neighbors. For bounded measurable $f:X\to \mathbb{R}$, set

$Z_{\mathrm{\Lambda }}^{h,\eta }:={\sum }_{\tau \in \{-1,+1{\}}^{\mathrm{\Lambda }}}{e}^{-\beta H_{\mathrm{\Lambda }}^h(\tau \mid \eta )},{\gamma }_{\mathrm{\Lambda }}^h(f\mid \eta ):=(Z_{\mathrm{\Lambda }}^{h,\eta }{)}^{-1}{\sum }_{\tau \in \{-1,+1{\}}^{\mathrm{\Lambda }}}f(\tau \eta )e^{-\beta H_{\mathrm{\Lambda }}^h(\tau \mid \eta )}\mathrm{.}$

Here $\tau \eta$ denotes the configuration equal to $\tau$ on $\mathrm{\Lambda }$ and to $\eta$ on ${\mathrm{\Lambda }}^c$.

For each field $h$, these kernels are positive normalized probability kernels. They are proper, meaning that they leave every event determined by the exterior spins fixed; consistent, meaning ${\gamma }_{\mathrm{\Lambda }}^h{\gamma }_{\mathrm{\Delta }}^h={\gamma }_{\mathrm{\Lambda }}^h$ whenever finite $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$; and finite range, meaning that for every finite $\mathrm{\Delta }$ and cylinder event $A$, $\eta \mapsto {\gamma }_{\mathrm{\Delta }}^h({\mathbf{1}}_A\mid \eta )$ depends on only finitely many exterior coordinates.

For $i\mathrm{\ne }j$, write ${\gamma }_i^h(\cdot \mid \eta ):={\gamma }_{\{i\}}^h(\cdot \mid \eta )$ for the one-site law and define the influence coefficient

$c_{ij}^h:={\sup }_{\eta ,{\eta }^{\mathrm{\prime }}:{\eta }_k={\eta }_k^{\mathrm{\prime }}(k\mathrm{\ne }j)}\frac{1}{2}{\sum }_{s\in \{-1,+1\}}\mathrm{\mid }{\gamma }_i^h(s\mid \eta )-{\gamma }_i^h(s\mid {\eta }^{\mathrm{\prime }})\mathrm{\mid },c_{ii}^h:=0.$

Then $c_{ij}^h\le \tanh (\beta \mathrm{\mid }J\mathrm{\mid })$ when $\parallel i-j{\parallel }_1=1$ and $c_{ij}^h=0$ otherwise. Consequently, if

$2d\tanh (\beta \mathrm{\mid }J\mathrm{\mid })<1,$

then every influence row sum is strictly below $1$. More generally, if $2d\tanh (\beta \mathrm{\mid }J\mathrm{\mid })<1$, then for any two site fields $h,\tilde{h}\in {\mathbb{R}}^{{\mathbb{Z}}^d}$ with the same $d,\beta ,J$, the matrix $C$ defined by $C_{ij}=\tanh (\beta \mathrm{\mid }J\mathrm{\mid })$ when $\parallel i-j{\parallel }_1=1$ and $C_{ij}=0$ otherwise is a common finite-range influence envelope with row sums below $1$. Thus, under this inequality, the finite-volume comparison estimate of the Dobrushin comparison theorem applies to ${\gamma }^h$ and ${\gamma }^{\tilde{h}}$. This is a sufficient criterion; failure of the inequality is not claimed to imply nonuniqueness.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Finite-spin Dobrushin influence matrix

Ising influence bounded by hyperbolic tangent

Finite-spin Dobrushin comparison and boundary response