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arXiv · 2207.13673

Multiscale coupling and the maximum of $\mathcal{P}(ϕ)_2$ models on the torus

Nikolay Barashkov·Trishen S. Gunaratnam·Michael Hofstetter
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Abstract

We establish a coupling between the $\mathcal{P}(ϕ)_2$ 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 $ϕ^4_2$ 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 $\mathcal{P}(ϕ)_2$. 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 $\mathcal{P}(ϕ)_2$ field on the discretised torus with mesh-size $ε> 0$ converges in distribution to a randomly shifted Gumbel distribution as $ε\rightarrow 0$.

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Nikolay Barashkov, Trishen S. Gunaratnam, Michael Hofstetter. 2023-11-10. Multiscale coupling and the maximum of $\mathcal{P}(ϕ)_2$ models on the torus. https://arxiv.org/abs/2207.13673

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