PhyslibAlpha.ProbabilisticTheory.LocalNet.Lattice

Given a group $G$ acting on a set of sites $S$, there exists an induced group action on the collection of finite regions of the lattice (represented as finite subsets of $S$, denoted $\text{Finset } S$). For any group element $g \in G$ and any finite region $\Lambda \subseteq S$, the translated region is defined as $g \cdot \Lambda = \{g \cdot s \mid s \in \Lambda\}$.

For a set of sites $S$, the separation relation on the collection of finite regions $\text{Finset } S$ is defined such that two regions $X$ and $Y$ are separated if and only if they are disjoint, denoted $X \cap Y = \emptyset$. This relation satisfies the properties of symmetry ($X \cap Y = \emptyset \implies Y \cap X = \emptyset$) and monotonicity ($X' \subseteq X, Y' \subseteq Y, X \cap Y = \emptyset \implies X' \cap Y' = \emptyset$).

Let $G$ be a group acting on a set of sites $S$. For any finite regions $X, Y \in \text{Finset } S$, let the separation relation be defined by disjointness, $X \cap Y = \emptyset$. Then the group action of $G$ on the regions preserves and reflects this separation: for any $g \in G$, $X \cap Y = \emptyset$ if and only if $g \cdot X \cap g \cdot Y = \emptyset$.

Given a finite region $X \subset S$ and a non-negative radius $r \in \mathbb{R}_{\ge 0}$, the enlargement of $X$ by $r$ is the finite set of sites defined by the union $\bigcup_{a \in X} N_r(a)$, where $N_r(a)$ denotes the neighborhood of site $a$ with radius $r$. This set contains all sites $c \in S$ that are within radius $r$ of at least one site in $X$.

For a non-negative radius $r \in \mathbb{R}_{\ge 0}$ and a finite region of sites $X \subset S$, a site $c \in S$ is an element of the enlargement of $X$ by $r$ if and only if there exists a site $a \in X$ such that $c$ is in the neighborhood of $a$ with radius $r$, denoted by $N_r(a)$.

This definition equips the collection of finite regions $\text{Finset } S$ (finite sets of sites) with an enlargement structure derived from the neighbor data $nb$. For a given radius $r \in \mathbb{R}_{\ge 0}$ and a finite region $X \subset S$, the enlargement $E(r, X)$ is defined as the union of neighborhoods of all sites in $X$: $$E(r, X) = \bigcup_{a \in X} N_r(a)$$ where $N_r(a)$ is the neighborhood of site $a$ with radius $r$. This enlargement operation satisfies the following properties: - **Zero radius**: $E(0, X) = X$, meaning enlargement by zero radius returns the original region. - **Region monotonicity**: If $X \subseteq Y$, then $E(r, X) \subseteq E(r, Y)$. - **Radius monotonicity**: If $r \le s$, then $E(r, X) \subseteq E(s, X)$. - **Radius subadditivity**: The composition of two enlargements is contained within the enlargement of the sum of their radii, i.e., $E(s, E(r, X)) \subseteq E(r+s, X)$.

Let $S$ be a set of sites and $G$ a group acting on $S$. A neighbor structure $nb$ is said to be equivariant if for every radius $r \in \mathbb{R}_{\ge 0}$, group element $g \in G$, and site $s \in S$, the neighborhood of the transformed site $g \cdot s$ at radius $r$ is equal to the neighborhood of $s$ transformed by $g$. That is, $nb(r, g \cdot s) = g \cdot nb(r, s)$, where the group action on the right-hand side is the lifted action of $G$ on the set of sites.

Let $S$ be a set of sites and $G$ a group acting on $S$. Let $N_r(s) \subseteq S$ denote the neighborhood of a site $s \in S$ with radius $r \in \mathbb{R}_{\ge 0}$. If the neighbor structure is equivariant, such that $N_r(g \cdot s) = g \cdot N_r(s)$ for all $g \in G$, $s \in S$, and $r \in \mathbb{R}_{\ge 0}$, then the enlargement of any finite region $X \subseteq S$ (defined as $E(r, X) = \bigcup_{a \in X} N_r(a)$) is also equivariant. That is, for any group element $g \in G$, radius $r \in \mathbb{R}_{\ge 0}$, and finite region $X$, the following holds: $$E(r, g \cdot X) = g \cdot E(r, X)$$