论文标题
关于吉布斯措施的评论与对数相关的高斯田地
A remark on Gibbs measures with log-correlated Gaussian fields
论文作者
论文摘要
我们在$ d $维圆环上使用对数相关的基本高斯田地研究吉布斯的测量。在偶然的情况下，纳尔逊的论点遵循了这种吉布斯措施的构建。在本文中，我们考虑了与四分之一相互作用的聚焦案例。使用变分公式，我们证明了Gibbs测量的不符合性。当$ d = 2 $时，我们的论点提供了一个替代的证明，证明了Brydges and Slade（1996）的焦点$φ^4_2 $的非正相关结果。此外，我们提供了确切的差异速率，其中常数的特征是特定伯恩斯坦在$ \ mathbb {r}^d $上的不等式的最佳常数。我们还通过立方相互作用进行了构造聚焦吉布斯测量。在附录中，我们提出了（a）Gibbs对二维Zakharov系统测量的非差异性，以及（b）使用更平滑的基础高斯措施来构建聚焦四分之一的吉布斯测量，显示了对数相关的Gibbs测量与聚焦四分之一相互作用的临界性质。
We study Gibbs measures with log-correlated base Gaussian fields on the $d$-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson's argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove non-normalizability of the Gibbs measure. When $d = 2$, our argument provides an alternative proof of the non-normalizability result for the focusing $Φ^4_2$-measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein's inequality on $\mathbb{R}^d$. We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) non-normalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.
