论文标题
正方形晶格上键渗透的随机矩阵表示的精确有限尺寸缩放
Exact finite-size scaling for the random-matrix representation of bond percolation on square lattice
论文作者
论文摘要
我们报告了在两个维度的正方形晶格上的键渗透模型的随机治疗方法，其职业概率$ p $。渗透问题映射到一个随机复杂矩阵中，该矩阵由两个随机的实价元素组成的元素$+1 $和$ -1 $，分别为$ p $和$ 1-p $。我们发现，由于前两个极限限制中前两个极端特征值的合并，可以通过幂律差异的出现来检测渗滤过渡的发作。我们开发了一种通用的有限尺寸缩放定律，该法律完全表征了极端特征值波动的缩放行为，从一组通用缩放指数和振幅方面。我们利用相对熵作为第一和第二大极端特征值的两个分布之间差异的索引，以表明其最小值是缩放框架的基础。我们的研究可能会为开发具有不同应用的新方法和算法在机器学习，复杂系统和统计物理学中的应用提供侵入。
We report on the exact treatment of a random-matrix representation of bond percolation model on a square lattice in two dimensions with occupation probability $p$. The percolation problem is mapped onto a random complex matrix composed of two random real-valued matrices of elements $+1$ and $-1$ with probability $p$ and $1-p$, respectively. We find that the onset of percolation transition can be detected by the emergence of power-law divergences due to the coalescence of the first two extreme eigenvalues in the thermodynamic limit. We develop a universal finite-size scaling law that fully characterizes the scaling behavior of the extreme eigenvalue's fluctuation in terms of a set of universal scaling exponents and amplitudes. We make use of the relative entropy as an index of the disparity between two distributions of the first and second-largest extreme eigenvalues, to show that its minimum underlies the scaling framework. Our study may provide an inroad for developing new methods and algorithms with diverse applications in machine learning, complex systems, and statistical physics.
