论文标题
多尺度耦合和$ \ MATHCAL {p}（ϕ）_2 $模型的最大值
Multiscale coupling and the maximum of $\mathcal{P}(ϕ)_2$ models on the torus
论文作者
论文摘要
我们在所有空间尺度上在二维单位圆环上的$ \ Mathcal {P}（P}（P}（ϕ）_2 $ MAUSE与高斯自由场之间建立耦合，通过差异字段上的概率规律性估计来量化。我们的结果包括研究良好的$ ϕ^4_2 $度量。该证明使用用于定义耦合的Polchinski重量化组方法与$ \ Mathcal {P}（P}（ϕ）_2 $的Boué-Dupuis随机控制表示形式）之间的确切对应关系。更确切地说，我们表明差异字段是从变异问题的特定最小化器中获得的。这允许将使用离散谐波分析工具获得的最小化器的小规模转移到差异字段中。作为耦合的应用，我们证明了$ \ Mathcal {p}（ϕ）_2 $字段的最大值，该_2 $字段在带有网格大小的$ε> 0 $的离散的圆环上，将分布收敛到随机移位的gumbel分布，为$ε\ rightarrow 0 $。
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 $ε\rightarrow 0$.
