Sasidevan V
Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Aug;88(2):022140. doi: 10.1103/PhysRevE.88.022140. Epub 2013 Aug 26.
We study the continuum percolation problem of overlapping disks with a distribution of radii having a power-law tail; the probability that a given disk has a radius between R and R+dR is proportional to R(-(a+1)), where a>2. We show that in the low-density nonpercolating phase, the two-point function shows a power-law decay with distance, even at arbitrarily low densities of the disks, unlike the exponential decay in the usual percolation problem. As in the problem of fluids with long-range interaction, we argue that in our problem, the critical exponents take their short-range values for a>3-η(sr) whereas they depend on a for a<3-η(sr) where η(sr) is the anomalous dimension for the usual percolation problem. The mean-field regime obtained in the fluid problem corresponds to the fully covered regime, a≤2, in the percolation problem. We propose an approximate renormalization scheme to determine the correlation length exponent ν and the percolation threshold. We carry out Monte Carlo simulations and determine the exponent ν as a function of a. The determined values of ν show that it is independent of the parameter a for a>3-η(sr) and is equal to that for the lattice percolation problem, whereas ν varies with a for 2<a<3-η(sr). We also determine the percolation threshold of the system as a function of the parameter a.
我们研究了半径分布具有幂律尾部的重叠圆盘的连续渗流问题；给定圆盘半径在(R)到(R + dR)之间的概率与(R^{-(a + 1)})成正比，其中(a > 2)。我们表明，在低密度非渗流相中，两点函数随距离呈现幂律衰减，即使在圆盘密度任意低的情况下也是如此，这与通常渗流问题中的指数衰减不同。正如在具有长程相互作用的流体问题中一样，我们认为在我们的问题中，对于(a > 3 - \eta_{(sr)})，临界指数取其短程值，而对于(a < 3 - \eta_{(sr)})，它们依赖于(a)，其中(\eta_{(sr)})是通常渗流问题中的反常维度。在流体问题中得到的平均场区域对应于渗流问题中完全覆盖的区域，即(a \leq 2)。我们提出了一种近似重整化方案来确定关联长度指数(\nu)和渗流阈值。我们进行了蒙特卡罗模拟，并确定了作为(a)的函数的指数(\nu)。所确定的(\nu)值表明，对于(a > 3 - \eta_{(sr)})，它与参数(a)无关，并且等于晶格渗流问题中的值，而对于(2 < a < 3 - \eta_{(sr)})，(\nu)随(a)变化。我们还确定了系统的渗流阈值作为参数(a)的函数。
