论文标题
离散高斯自由场的梯度平方的属性
Properties of the gradient squared of the discrete Gaussian free field
论文作者
论文摘要
在本文中，我们研究了$u_ε= u/ε\ cap \ cap \ mathbb {z}^d $，$ u \ subset \ subset \ mathbb {r}^d $ and $ d \ geq 2 $。该场的协方差结构是转移电流矩阵的函数，这将模型与一类系统（例如，二聚体模型中的Abelian Sandpile模型的高度 - 一个场）相关联，该模型由于传输电流的快速衰减而具有高斯极限。确实，我们证明，正确重新缩放的场在适当的本地Besov-Hölder空间中汇聚为白噪声。此外，在不同的重新缩放下，我们确定$ k $ - 点相关功能和$u_ε$上的累积函数，在连续限制中为$ε\至0 $。该结果与Abelian Sandpile（\ citet {durRe}）的高度一体字段的模拟限制有关，并在$ d = 2 $中具有相同的共同协变属性。
In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in $U_ε=U/ε\cap \mathbb{Z}^d$, $U\subset \mathbb{R}^d$ and $d\geq 2$. The covariance structure of the field is a function of the transfer current matrix and this relates the model to a class of systems (e.g. height-one field of the Abelian sandpile model or pattern fields in dimer models) that have a Gaussian limit due to the rapid decay of the transfer current. Indeed, we prove that the properly rescaled field converges to white noise in an appropriate local Besov-Hölder space. Moreover, under a different rescaling, we determine the $k$-point correlation function and cumulants on $U_ε$ and in the continuum limit as $ε\to 0$. This result is related to the analogue limit for the height-one field of the Abelian sandpile (\citet{durre}), with the same conformally covariant property in $d=2$.
