We revisit a known model in which (conducting) blocks are hierarchically and randomly deposited on a $D$-dimensional substrate according to a hyperbolic size law with the block size decreasing by a factor $\lambda >1$ in each subsequent generation. In the first part of the paper the number of coastal points (in $D=1$) or coastline segments (in $D=2$) is calculated, which are points or lines that separate a region at “sea level” and an elevated region. We find that this number possesses a non-universal character, implying a Euclidean geometry below a threshold value $P_c$ of the deposition probability $P$, and a fractal geometry above this value. Exactly at the threshold, the geometry is logarithmic fractal. The number of coastline segments in $D=2$ turns out to be exactly twice the number of coastal points in $D=1$. We comment briefly on the surface morphology and derive a roughness exponent $\alpha$. In the second part, we study the percolation probability for a current in this model and two extensions of it, in which both the scale factor and the deposition probability can take on different values between generations. We find that the percolation threshold $P_c$ is located at exactly the same value for the deposition probability as the threshold probability of the number of coastal points. This coincidence suggests that exactly at the onset of percolation for a conducting path, the number of coastal points exhibits logarithmic fractal behaviour.

我们重新审视一个已知的模型，在该模型中，（导电）块被分层且随机地存放在一个 $d$块尺寸减小一倍的双曲线尺寸定律的三维衬底 $\lambda >\mathrm{1个}$在随后的每一代中。在本文的第一部分中，沿海点的数量（以$d=\mathrm{1个}$）或海岸线段（在 $d=\mathrm{2个}$）被计算出来，这些点或线将“海平面”上的区域与高架区域分开。我们发现该数字具有非通用特征，这意味着欧几里德几何形状低于阈值$P_C$ 沉积概率 $P$，以及高于该值的分形几何形状。恰好在阈值处，几何形状是对数分形。海岸线段数$d=\mathrm{2个}$ 原来是海岸点数量的两倍 $d=\mathrm{1个}$。我们简要评论表面形态并得出粗糙度指数$\alpha$。在第二部分中，我们研究了该模型中电流的渗流概率及其两个扩展，其中比例因子和沉积概率在几代之间可以取不同的值。我们发现渗滤阈值$P_C$沉积概率的值与沿海点的数量的阈值概率完全相同。这种巧合表明，恰好在导电路径渗流开始时，沿海点的数量呈现出对数分形行为。