论文标题
自由费点处的两个周期性加权多米诺骨牌和正弦 - 戈登场：i
Two-periodic weighted dominos and the sine-Gordon field at the free fermion point: I
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
In this paper we investigate the height field of a dimer model/random domino tiling on the plane at a smooth-rough (i.e. gas-liquid) transition. We prove that the height field at this transition has two-point correlation functions which limit to those of the massless sine-Gordon field at the free fermion point, with parameters $(4π, z)$ where $z\in \mathbb{R}\setminus \{0\}$. The dimer model is on $ε\mathbb{Z}^2$ and has a two-periodic weight structure with weights equal to either 1 or $a=1-C|z|ε$, for $0<ε$ small (tending to zero). In order to obtain this result, we provide a direct asymptotic analysis of a double contour integral formula of the correlation kernel of the dimer model found by Fourier analysis. The limiting field interpolates between the Gaussian free field and white noise and the main result gives an explicit connection between tiling/dimer models and the law of a two-dimensional non-Gaussian field.
