Explosive percolation (EP) has received significant research attention due to its rich and anomalous phenomena near criticality. In our recent study [], we demonstrated that the correct critical behaviors of EP in infinite dimensions (complete graph) can be accurately extracted using the event-based method, with finite-size scaling behaviors still described by the standard finite-size scaling theory. We perform an extensive simulation of EPs on hypercubic lattices ranging from dimensions d = 2 to 6, and find that the critical behaviors consistently obey the standard finite-size scaling theory. Consequently, we obtain a high-precision determination of the percolation thresholds and critical exponents, revealing that EPs governed by the product and sum rules belong to different universality classes. Remarkably, despite the mean of the dynamic pseudocritical point T L deviating from the infinite-lattice criticality by a distance determined by the d -dependent correlation-length exponent, T L follows a normal (Gaussian) distribution across all dimensions, with a standard deviation proportional to 1 / V , where V denotes the system volume. A theoretical argument associated with the central-limit theorem is further proposed to understand the probability distribution of T L . These findings offer a comprehensive understanding of critical behaviors in EPs across various dimensions, revealing a different dimension-dependence compared to standard bond percolation. Published by the American Physical Society 2024
Explosive percolation (EP) has received significant research attention due to its rich and anomalous phenomena near criticality. In our recent study [], we demonstrated that the correct critical behaviors of EP in infinite dimensions (complete graph) can be accurately extracted using the event-based method, with finite-size scaling behaviors still described by the standard finite-size scaling theory. We perform an extensive simulation of EPs on hypercubic lattices ranging from dimensions d = 2 to 6, and find that the critical behaviors consistently obey the standard finite-size scaling theory. Consequently, we obtain a high-precision determination of the percolation thresholds and critical exponents, revealing that EPs governed by the product and sum rules belong to different universality classes. Remarkably, despite the mean of the dynamic pseudocritical point T L deviating from the infinite-lattice criticality by a distance determined by the d -dependent correlation-length exponent, T L follows a normal (Gaussian) distribution across all dimensions, with a standard deviation proportional to 1 / V , where V denotes the system volume. A theoretical argument associated with the central-limit theorem is further proposed to understand the probability distribution of T L . These findings offer a comprehensive understanding of critical behaviors in EPs across various dimensions, revealing a different dimension-dependence compared to standard bond percolation. Published by the American Physical Society 2024
Abstract Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.
Finite-size scaling (FSS) for a critical phase transition (t=0) states that within a window of size ∣t∣∼L−1/ν, the scaling behavior of any observable Q in a system of linear size L asymptotically follows a scaling form as Q(t,L)=LYQQ~(tL1/ν), where ν is the correlation-length exponent, YQ is an FSS exponent and Q~(x) is a function of the scaled distance-to-criticality x≡tL1/ν. We systematically study the asymptotic scaling behavior of Q~(∣x∣→∞) for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with t→0 and ∣x∣→∞. This crossover FSS theory predicts that when the criticality is approached at a slower speed as ∣t∣∼L−λ with λ<1/ν, the FSS becomes λ-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality t=0 can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.
We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512, and estimate the percolation thresholds to be pc(bond)=0.24881182(10) and pc(site)=0.3116077(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent dmin=1.3756(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.
We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities p(r)∝1/r2+σ, establishing the universality diagram in the long-range (LR) regime for σ≤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<σ≤2), LR Wilson--Fisher B (2/3<σ≤1), and LR mean-field (MF) (0<σ≤2/3). 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≃3/2 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.
Abstract We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions d c . Behaviour at the critical point is non-universal in d > d c = 6 dimensions. Proliferation of the largest clusters, with fractal dimension 4, is associated with the breakdown of hyperscaling there when free boundary conditions are used. But when the boundary conditions are periodic, the maximal clusters have dimension D = 2 d /3, and obey random-graph asymptotics. Universality is instead manifested at the pseudocritical point, where the failure of hyperscaling in its traditional form is universally associated with random-graph-type asymptotics for critical cluster sizes, independent of boundary conditions.
Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability Ep, the percolation probability P, and the probability Wn for the appearance of n percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions L. Using the HMCSM, we calculate the existence probability Ep and the percolation probability P for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of Ep and P in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that Ep and P have universal finite-size scaling functions. This implies that the critical Ep is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that Wn for site and bond percolation on sc lattices have universal finite-size scaling functions.