Area law for three-dimensional U(1) lattice gauge theory with the Wilson action at every coupling

The results of version 3 are unchanged, and every theorem, lemma, proposition and equation keeps its number. The abstract is new and the introduction revised; the former Part S3 is dropped, and its numerical checks are described in the README in scripts.zip; Tables S1.3 and S1.4 are written out in Section S1.6 and Remark S1.1; and Part S4 is shortened where it re-proved a standard formula, gave a second proof or made a point twice, its introduction saying in a table which of its sections completes which part of the main text. Version 4 corrects the following statements of version 3; the arguments of the proofs are unchanged.

Statements that a reader may have quoted or used, in the order of the paper:

• Abstract: for the Villain action, Göpfert and Mack proved that a string tension defined by an iterated limit is positive at every coupling; version 3 credited them with the area law uniform in the box, which is stronger.
• Section 1.2: through Fröhlich's comparison, an area law at every coupling for the closed subgroups of U(n) containing the scalars requires the Wilson case of U(1); version 3 stated this for groups with a central U(1), without restricting it to that comparison.
• Section 5, first sentence: the probability is at most 2^(4|C|+5) e^(−RT), as in Theorem 5.1, not e^(−RT).
• Section 8.1, opening: the min-cut lower tail bounds the exponential moment of the area deficit, and the averaged bound is e^(−σ_V(1−η_S)RT + c_N(R+T)) · E_D e^(σ_V(RT − A_{G′}(C))); version 3 equated the two and wrote e^(−σRT) · E_D e^(σ(RT − A)) with an undefined σ.
• Remark 8.1: the claim, made without proof, that the right side of Theorem 2.5 is not monotone in β is withdrawn; the remark now says that raising β raises τ and lowers δ, which push that side in opposite directions.
• Section 8.2, end: the constants fixed before K depend on τ at most through m̃, the scales of item 11 of the ordered list (Section S1.1.1) and c_dep; version 3 said that they do not depend on τ.
• Proof of Proposition 9.1: the remark of Messager, Miracle-Solé and Pfister (p. 21) states the bound for every α; version 3 said that it is stated only for α equal to the phase of the interaction.
• Introduction of Appendix E: Proposition E.10 is stated with its hypothesis, that a character coefficient of the weight exceeds that of the Villain weight ϑ_t; version 3 stated it without this hypothesis.
• Remark E.9(iii): the claim that the non-monotonicity of Theorem E.8 needs light tails and large β or t is withdrawn; the remark now says that it comes from them.
• Section S1.1.4(d): the conditions of Section 8 are η_T + η_S ≤ 1/2 and σ_V ≤ 1, as Section 8.1 already said; version 3 listed only the first, and β₀(K) is unchanged.

Descriptions of the argument:

• Section 1.3, step (ii): the cut functional of the regularization penalizes leaving blocks crowded with marks unfilled, not the crowded blocks themselves, and the constants of the output are bounded independently of β, K, C and D; version 3 said that they are independent of them.
• Section 1.3, step (iii): on each output the loop decays exponentially in the min-cut area; version 3 wrote "like the exponential of" it, here and in its abstract (the new abstract omits this step).
• Proof of Theorem S4.8.36, step (K1): of the two bounds, only this one rests on positivity; version 3 called it the one place where the proof needs positivity, but step 4 uses it too.