Maximum and coupling of the sine-Gordon field

Abstract

For 0 < β < 6π, we prove that the distribution of the centred maximum of the ε-regularised continuum sine-Gordon field on the two-dimensional torus converges to a randomly shifted Gumbel distribution as ε → 0. Our proof relies on a strong coupling at all scales of the sine-Gordon field with the Gaussian free field, of independent interest, and extensions of existing methods for the maximum of the lattice Gaussian free field.

Elliptic stochastic quantization of Sinh-Gordon QFT

“…Let us remark that, although both the elliptic and parabolic SQ methods so far often fall short of providing enough information on the decay of correlation functions of ν, they are capable of proving pathwise properties even when absolute continuity is lost, since (5) and (6) provide a coupling between the Gaussian Free Field (having law µ) and the interacting fields (having law ν). Similar results have also been obtained using the Polchinski equation, see, e.g., [17].…”

“…Proof The proof follows from the fact that we can identify S ′ (R 2 , S ′ (εZ 2 )) with S ′ (R 2 × εZ 2 ), from the monotone convergence theorem and from the definition of the norm (17) of Besov space B s1 p,p,ℓ1 (εZ 2 ). ✷ We now propose an extension to the Besov spaces on R 2 × εZ 2 of the results of the previous sections.…”

The Coleman correspondence at the free fermion point

“…Even in finite volume, the ultraviolet (short-distance) stability of both models is delicate when β 4π (for β < 4π the situation is simpler). For example, various constructions of the sine-Gordon model exist under different assumptions (see in particular [3,6,9,16,17,20,26,36,38,43]), but none of these covers all β ∈ (0, 8π) and all z ∈ R, even for interactions in a fixed finite volume. The analysis of the infinite volume limit is a different problem.…”