Finite-size scaling of energylike quantities in percolation

Abstract

We study the bond-percolation model in two and three dimensions by Monte Carlo simulation, and investigate the finite-size scaling behavior of several quantities that account for fluctuations of the total numbers of clusters and occupied bonds, ${N}_{c}$ and ${N}_{b}$, respectively. These quantities include ${C}_{2c}=(⟨{N}_{c}^{2}⟩\ensuremath{-}{⟨{N}_{c}⟩}^{2})∕{L}^{d}$ and ${C}_{cb}=(⟨{N}_{c}{N}_{b}⟩\ensuremath{-}⟨{N}_{c}⟩⟨{N}_{b}⟩)∕{L}^{d}$, where $L$ is the linear system size and $d$ is the spatial dimensionality. In statistical models with thermal fluctuations, ${C}_{2c}$ and ${C}_{cb}$ are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of ${C}_{2c}$ and ${C}_{cb}$ is described by the thermal critical exponent ${y}_{t}\text{\ensuremath{-}}d$. We also measure quantity ${\ensuremath{\kappa}}_{b}=2⟨{N}_{b}{S}_{2}⟩∕⟨{S}_{2}⟩\ensuremath{-}⟨{N}_{b}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩∕⟨{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩\ensuremath{-}⟨{N}_{b}⟩$ and an analogous quantity ${\ensuremath{\kappa}}_{c}$ for ${N}_{c}$, where ${S}_{2}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}$ are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ diverge as ${L}^{{y}_{t}}$ for $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The analysis of the data of ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ yields ${y}_{t}=1.145(2)$ for the three-dimensional percolation, in good agreement with existing results.