The area law is the criterion for confinement in a gauge theory: the expectation value of a large Wilson loop falls off exponentially with the area it encloses, rather than its perimeter.

For a planar loop \(C\) enclosing area \(A\),

\[ \langle W(C)\rangle \sim \exp\!\big(-\sigma\,A\big)\qquad(\text{confinement}), \]

with \(\sigma\) the string tension. The competing behaviour,

\[ \langle W(C)\rangle \sim \exp\!\big(-\mu\,P\big)\qquad(\text{deconfinement / screening}), \]

depends only on the perimeter \(P\) and indicates free or screened charges.

An area law is equivalent to a linearly rising static potential. Writing a temporal \(R\times \mathcal{T}\) loop, \(\langle W\rangle\sim e^{-V(R)\mathcal{T}}\), the area law \(e^{-\sigma R\mathcal{T}}\) gives

\[ V(R)=\sigma R , \]

the flux-tube potential between static quarks. Separating the charges costs energy growing without bound — they are confined into bound states.

At finite temperature the two Wilson-loop orientations behave differently:

• temporal loops lose the area law above \(T_c\) (\(\sigma(T)\to0\)): electric deconfinement;
• spatial loops keep an area law at all \(T\), defining the spatial string tension \(\sigma_s(T)\), which is governed by the 3D magnetic theory MQCD.

The area-law slope of spatial loops is exactly what is fitted on the lattice to test dimensional reduction.