We consider discrete random fractal surfaces with negative Hurst exponent $H<0$. A random colouring of the lattice is provided by activating the sites at which the surface height is greater than a given level $h$. The set of activated sites is usually denoted as the excursion set. The connected components of this set, the level clusters, define a one-parameter ($H$) family of percolation models with long-range correlation in the site occupation. The level clusters percolate at a finite value $h=h_c$ and for $H\leq-\frac{3}{4}$ the phase transition is expected to remain in the same universality class of the pure (i.e. uncorrelated) percolation. For $-\frac{3}{4}<H< 0$ instead, there is a line of critical points with continously varying exponents. The universality class of these points, in particular concerning the conformal invariance of the level clusters, is poorly understood. By combining the Conformal Field Theory and the numerical approach, we provide new insights on these phases. We focus on the connectivity function, defined as the probability that two sites belong to the same level cluster. In our simulations, the surfaces are defined on a lattice torus of size $M\times N$. We show that the topological effects on the connectivity function make manifest the conformal invariance for all the critical line $H<0$. In particular, exploiting the anisotropy of the rectangular torus ($M\neq N$), we directly test the presence of the two components of the traceless stress-energy tensor. Moreover, we probe the spectrum and the structure constants of the underlying Conformal Field Theory. Finally, we observed that the corrections to the scaling clearly point out a breaking of integrability moving from the pure percolation point to the long-range correlated one.

中文翻译：

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远程相关随机曲面的拓扑效应和共形不变性

我们考虑具有负Hurst指数$ H <0 $的离散随机分形曲面。通过激活表面高度大于给定水平$ h $的位置，可以对晶格进行随机着色。激活位点集通常表示为偏移集。该集合的连接组件，即水平簇，定义了一个渗滤模型族的一参数（$ H $）系列，在场地占用中具有长期相关性。级别簇以有限值$ h = h_c $渗透，并且对于$ H \ leq- \ frac {3} {4} $，相变预计将保持在纯（即不相关）渗透的同一通用类中。相反，对于$-\ frac {3} {4} <H <0 $，存在一系列临界点，它们的指数不断变化。这些要点的通用性 尤其是关于水平簇的共形不变性的了解很少。通过整合共形场理论和数值方法，我们提供了关于这些阶段的新见解。我们专注于连接功能，定义为两个站点属于同一级别群集的概率。在我们的仿真中，曲面定义在大小为$ M \ N $的晶格圆环上。我们表明，对连通性函数的拓扑效应使所有关键线$ H <0 $都表现出共形不变性。特别是，利用矩形环面的各向异性（$ M \ neq N $），我们直接测试了无痕应力能张量的两个分量的存在。此外，我们探讨了潜在的共形场理论的光谱和结构常数。最后，