Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime

Abstract

We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+\sigma }$, establishing the universality diagram in the long-range (LR) regime for $\sigma \le 2$. 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 ($1<\sigma \le 2$), LR Wilson--Fisher B ($2/3<\sigma \le 1$), and LR mean-field (MF) ($0<\sigma \le 2/3$). In the LR-WF-B regime, the anomalous dimension is consistent with $\eta =2-\sigma$, in agreement with mathematical results for $2/3<\sigma <1$, while the correlation-length exponent $\nu (\sigma )$ exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although $\eta$ remains close to $2-\sigma$ for smaller $\sigma$, statistically resolvable deviations $\delta \eta (\sigma )=\eta -(2-\sigma )>0$ start to appear near $\sigma \simeq 3/2$ and grow toward the short-range crossover at $\sigma =2$. 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.

14 pages, 7 figures