jats:titleAbstract</jats:title>jats:pWe establish a coupling between the jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {P}}(\phi )_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miP</mml:mi> mml:msub mml:mrow mml:mo(</mml:mo> mml:miϕ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mn2</mml:mn> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result includes the well-studied jats:inline-formulajats:alternativesjats:tex-math$$\phi ^4_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msubsup mml:miϕ</mml:mi> mml:mn2</mml:mn> mml:mn4</mml:mn> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> measure. The proof uses an exact correspondence between the Polchinski renormalisation group approach, which is used to define the coupling, and the Boué–Dupuis stochastic control representation for jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {P}}(\phi )_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miP</mml:mi> mml:msub mml:mrow mml:mo(</mml:mo> mml:miϕ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mn2</mml:mn> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. More precisely, we show that the difference field is obtained from a specific minimiser of the variational problem. This allows to transfer regularity estimates for the small-scales of minimisers, obtained using discrete harmonic analysis tools, to the difference field.As an application of the coupling, we prove that the maximum of the jats:inline-formulajats:alternativesjats:tex-math$${\mathcal {P}}(\phi )_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miP</mml:mi> mml:msub mml:mrow mml:mo(</mml:mo> mml:miϕ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mn2</mml:mn> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> field on the discretised torus with mesh-size jats:inline-formulajats:alternativesjats:tex-math$$\epsilon > 0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miϵ</mml:mi> mml:mo></mml:mo> mml:mn0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> converges in distribution to a randomly shifted Gumbel distribution as jats:inline-formulajats:alternativesjats:tex-math$$\epsilon \rightarrow 0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miϵ</mml:mi> mml:mo→</mml:mo> mml:mn0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>