Theorem. The exact pure-gauge torus law [fdip-006R]
✍️source

  • November 1, 2026

For the square torus, take \(G=\{1,-1\}\) in its one-dimensional sign representation and use the counting convention. With \(n=|V|=P\) and \(\beta \ge 0\), \[\begin {aligned} \mathcal Z(\beta ) &=2^{n+1}\,\frac {(2\cosh \beta )^P+(2\sinh \beta )^P}{2},\\ Z_W(\beta )&=e^{-\beta P}\mathcal Z(\beta ). \end {aligned}\] Let \(C\) be a simple contractible edge loop bounding a specified disk of \(A\) square faces, where \(0<A<P\). Put \(t=\tanh \beta \). Its Wilson loop is \(W(C)=U_C\in \{1,-1\}\), and \[\langle W(C)\rangle _W =\frac {t^A+t^{P-A}}{1+t^P}.\] The finite formula includes the complementary region through the second term.

Proof. 

  • November 1, 2026

Write \(u_p=U_p\). By the plaquette counting lemma, every pattern with \(\prod _pu_p=1\) has \(2^{n+1}\) preimages. The indicator of this constraint is \((1+\prod _pu_p)/2\). Therefore \[\begin {aligned} \mathcal Z(\beta ) &=\frac {2^{n+1}}2\sum _{u\in \{1,-1\}^{\mathcal P}} \left (1+\prod _pu_p\right )\prod _p e^{\beta u_p}\\ &=\frac {2^{n+1}}2 \left [(2\cosh \beta )^P+(2\sinh \beta )^P\right ]. \end {aligned}\] Here each unweighted one-sign sum is \(2\cosh \beta \) and each sign-weighted sum is \(2\sinh \beta \). The displayed Wilson action constant gives the formula for \(Z_W\).

Multiplying the plaquette signs inside the chosen disk cancels every interior edge twice and leaves exactly the boundary product, so \(W(C)=\prod _{p\text { inside }C}u_p\). Insert this product into the preceding constrained sum. The two terms in the unnormalized numerator, after removing the common factor \(2^{n+1}/2\), are \[\begin {aligned} &(2\sinh \beta )^A(2\cosh \beta )^{P-A},\\ &(2\cosh \beta )^A(2\sinh \beta )^{P-A}. \end {aligned}\] Divide by the corresponding partition function and then by \((2\cosh \beta )^P\). The common action constant cancels, proving the expectation formula.

Fix \(\beta >0\) and a finite disk loop while both torus periods grow, so \(A\) stays fixed and \(P\to \infty \). Then \(0<t<1\), giving \[\begin {aligned} \lim _{P\to \infty }\langle W(C)\rangle _W &=t^A=e^{-\sigma A},\\ \sigma &=-\log (\tanh \beta )>0. \end {aligned}\] This is an exact area law for this two-dimensional pure-gauge limit. It is not a claim for a matter-coupled law or for higher-dimensional lattices. At \(\beta =0\), the finite formula instead gives zero for these nonempty proper disk loops, and \(Z_W(0)=2^{2n}\).