Finite box in the lattice

A finite cube-shaped subset of the integer lattice used as a finite region.

Finite box in the lattice

A finite box (or finite cube) in the lattice lattice-zd is a finite region of the form

\Lambda_L := \{x=(x_1,\dots,x_d)\in\mathbb{Z}^d : |x_i|\le L \text{ for all } i\},

where $L$ is a nonnegative natural-numbers .

This is the cube centered at the origin with side length $2L+1$ (in lattice units). Its cardinality is

|\Lambda_L|=(2L+1)^d.

Translations and other conventions.

Any translate $\Lambda_L+a:=\{x+a:x\in\Lambda_L\}$ is also a finite box.
Some authors use $\{0,1,\dots,L-1\}^d$ as the “box of side length $L$ ”; this differs from $\Lambda_L$ by translation and a minor change of parameter.

Finite boxes are common choices of “finite volume” regions for lattice models; associated notions of boundary-finite-region describe sites or edges near the complement.