The Largest Cluster in Subcritical Percolation

Abstract

The statistical behavior of the size (or mass) of the largest cluster in subcritical percolation on a finite lattice of size $N$ is investigated (below the upper critical dimension, presumably $d_c=6$). It is argued that as $N\to \infty$ the cumulative distribution function converges to the Fisher-Tippett (or Gumbel) distribution $e^{-e^{-z}}$ in a certain weak sense (when suitably normalized). The mean grows like $s_{\xi }^{\ast }\log N$, where $s_{\xi }^{\ast }(p)$ is a ``crossover size''. The standard deviation is bounded near $s_{\xi }^{\ast }\pi /\sqrt{6}$ with persistent fluctuations due to discreteness. These predictions are verified by Monte Carlo simulations on $d=2$ square lattices of up to 30 million sites, which also reveal finite-size scaling. The results are explained in terms of a flow in the space of probability distributions as $N\to \infty$. The subcritical segment of the physical manifold ($0<p<p_c$) approaches a line of limit cycles where the flow is approximately described by a ``renormalization group'' from the classical theory of extreme order statistics.

16 pages, 5 figs, expanded version to appear in Phys Rev E

Cited by in corpus (27)

• A fast Monte Carlo algorithm for site or bond percolation
• Thresholds for topological codes in the presence of loss
• Electrochemical kinetics of SEI growth on carbon black, II: Modeling
• Error Correction and Degeneracy in Surface Codes Suffering Loss
• Finite-size scaling in extreme statistics
• Directed percolation describes lifetime and growth of turbulent puffs and slugs
• Turbulence as a problem in non-equilibrium statistical mechanics
• Phase transition of computational power in the resource states for one-way quantum computation
• Fault-tolerance thresholds for the surface code with fabrication errors
• Universal measurement-based quantum computation with spin-2 Affleck-Kennedy-Lieb-Tasaki states
• An elementary renormalization-group approach to the Generalized Central Limit Theorem and Extreme Value Distributions
• Critical percolation in the dynamics of the 2d ferromagnetic Ising model
• Transitions in the computational power of thermal states for measurement-based quantum computation
• Classical Simulation of Quantum Error Correction in a Fibonacci Anyon Code
• Cluster size distributions in particle systems with asymmetric dynamics
• Nature of largest cluster size distribution at the percolation threshold
• Nuclear multifragmentation time-scale and fluctuations of largest fragment size
• Chemical fracture and distribution of extreme values
• Stochastic Renormalization Group in Percolation: I. Fluctuations and Crossover
• Order statistics of 1/f^α signals
• Critical behavior of the long-range Ising chain from the largest-cluster probability distribution
• Random Sequential Renormalization and Agglomerative Percolation in Networks: Application to Erd"os-R'enyi and Scale-free Graphs
• Phase separation and critical percolation in bidimensional spin-exchange models
• Non-Pauli errors can be efficiently sampled in qudit surface codes
• Records in the classical and quantum standard map
• Percolation in quantum computation and communication
• Asymptotic Behaviors of The Size of The Largest Cluster in One Dimensional Percolation