Abstract
AbstractWe establish a coupling between the $${\mathcal {P}}(\phi )_2$$
P
(
ϕ
)
2
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 $$\phi ^4_2$$
ϕ
2
4
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 $${\mathcal {P}}(\phi )_2$$
P
(
ϕ
)
2
. 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 $${\mathcal {P}}(\phi )_2$$
P
(
ϕ
)
2
field on the discretised torus with mesh-size $$\epsilon > 0$$
ϵ
>
0
converges in distribution to a randomly shifted Gumbel distribution as $$\epsilon \rightarrow 0$$
ϵ
→
0
.
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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