具有代数衰减相互作用的二维渗流 II：长程区域中的临界指数
Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime
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中文总结 AI 辅助
本文通过蒙特卡罗模拟明确了长程渗流的三个普适性区域，确定了各区域的临界指数特征，揭示了长程平均场区的渐近行为共存现象，为长程统计系统提供了关键基准。
中文摘要 AI 辅助
我们对具有代数衰减连接概率 $p(r)\propto 1/r^{2+\sigma}$ 的二维键渗流开展了全面的蒙特卡罗研究，确定了 $\sigma\le2$ 时长程（LR）区域的普适性相图。采用基于事件的系综方法，我们模拟了线性尺寸达 $L=16384$ 的系统，并研究了三个普适性区域：长程威尔逊-费舍尔（WF）A 区（$1<\sigma\le2$）、长程威尔逊-费舍尔 B 区（$2/3<\sigma\le1$）和长程平均场（MF）区（$0<\sigma\le2/3$）。在长程 WF-B 区，反常维度与 $\eta=2-\sigma$ 一致，这与 $2/3<\sigma<1$ 的数学结果相符，而关联长度指数 $\nu(\sigma)$ 呈现非平凡、非高斯的变化。在长程 WF-A 区，尽管较小 $\sigma$ 下 $\eta$ 仍接近 $2-\sigma$，但在 $\sigma\simeq3/2$ 附近开始出现统计可分辨的偏差 $\delta\eta(\sigma)=\eta-(2-\sigma)>0$，并向 $\sigma=2$ 的短程交叉处增大。最后，通过将基于事件的模拟与传统系综模拟相结合，我们揭示了长程 MF 区中完全图渐近行为与长程高斯不动点标度的共存。这些结果进一步阐明了长程渗流的临界性质，并为长程统计系统提供了关键基准。
英文摘要
We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+σ}$, establishing the universality diagram in the long-range (LR) regime for $σ\le2$. Using the event-based ensemble method, we simulate systems with linear sizes up to $L=16384$ and investigate three universality regimes: LR Wilson--Fisher (WF) A ($1<σ\le2$), LR Wilson--Fisher B ($2/3<σ\le1$), and LR mean-field (MF) ($0<σ\le2/3$). In the LR-WF-B regime, the anomalous dimension is consistent with $η=2-σ$, in agreement with mathematical results for $2/3<σ<1$, while the correlation-length exponent $ν(σ)$ exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although $η$ remains close to $2-σ$ for smaller $σ$, statistically resolvable deviations $δη(σ)=η-(2-σ)>0$ start to appear near $σ\simeq3/2$ and grow toward the short-range crossover at $σ=2$. Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.
