Finite box in the lattice

A finite box (or finite cube) in the integer lattice is a 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 integer.

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

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

Remarks

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 finite-volume regions for lattice models. The boundary of a finite region describes the sites or edges adjacent to the complement.