The suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent 1/φ(t)=1.8 ± 0.1.
The suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent 1/φ(t)=1.8 ± 0.1.
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.
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
A sudden change in size, number, or extent is rare, yet highly desired effect in theoretical models. The abrupt emergence of large-scale connectivity in a percolating network near the transition point gained the name explosive percolation. Classical network science lacks a simple recipe for explosiveness. Quantum networks governed by the laws of quantum mechanics intrinsically possess such an explosive component due to quantum superposition and measurements, which may be employed to achieve macroscopic explosiveness. Any percolating quantum network may exhibit explosive properties due to different state preparation and observation strategies near the transition point.
The invaded cluster algorithm [1] is generalized [2] to the tricritical point on the example of 2d Potts model with annealed dilution. Self-regulating procedure that locates the tricritical point in the two-parameter space spanned by temperature and chemical potential of vacancies is constructed based on geometrical arguments. The tricritical point is identified as a simultaneous percolation of the Fortuin-Kastelyn cluster and the geometrical cluster consisting of vacancies and isolated spins. The tricritical values of parameters and concentration are presented for q=1, 2, 3 and found to be in a good agreement with the best known results [3]. Scaling properties of the percolating scaling cluster and related critical exponents are also derived. Based on the idea that effective correlation of vacancies is important at the tricritical point, we also examine alternative stopping rules within generalized IC algorithm, and possible extension to higher dimensions. [1] J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, L. V. Chayes, Phys. Rev. Lett. 75, 2792 (1995) ; J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, L. M. Chayes, Phys. Rev. E 54, 1332 (1996). [2] I. Balog, K.Uzelac, preprint cond-mat/0703759. [3] X. Qian, Y. Deng, H. W. J. Bl¨ote, Phys. Rev. E 72, 056132 (2005).
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.
We examine the average cluster distribution as a function of lattice probability for a very small (L=6) lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of L=6 and L=63 lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.