In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r^{d+σ}, where r is the distance length between distinct sites and d=1. We introduce and test an order-N Monte Carlo algorithm and we determine as a function of σ the critical value C_{c} at which percolation occurs. The critical exponents in the range 0<σ<1 are reported. Our analysis is in agreement, up to a numerical precision ≈10^{-3}, with the mean-field result for the anomalous dimension η=2-σ, showing that there is no correction to η due to correlation effects. The obtained values for C_{c} are compared with a known exact bound, while the critical exponent ν is compared with results from mean-field theory, from an expansion around the point σ=1 and from the ɛ-expansion used with the introduction of a suitably defined effective dimension d_{eff} relating the long-range model with a short-range one in dimension d_{eff}. We finally present a formulation of our algorithm for bond percolation on general graphs, with order N efficiency on a large class of graphs including short-range percolation and translationally invariant long-range models in any spatial dimension d with σ>0.
We investigate the influence of the range of interactions in the two-dimensional bond percolation model by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number
z of equivalent neighbors. We also consider the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite
z are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for
q=3 and
q=4 Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as ${y}_{r}\ensuremath{\approx}2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.
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.
Dimensional correspondences have a long history in critical phenomena. Here, we review the effective dimension approach, which relates the scaling exponents of a critical system in d spatial dimensions with power-law decaying interactions r^{d+σ} to a local system, i.e., with finite-range interactions, in an effective fractal dimension d_{eff}. This method simplifies the study of long-range models by leveraging known results from their local counterparts. While the validity of this approximation beyond the mean-field level has been long debated, we demonstrate that the effective dimension approach, while approximate for non-Gaussian fixed points, accurately estimates the critical exponents of long-range models with an accuracy typically larger than 97%. To do so, we review perturbative renormalization-group (RG) results, extend the approximation's validity using functional RG techniques, and compare our findings with precise numerical data from conformal bootstrap for the two-dimensional Ising model with long-range interactions.
We investigate equivalent-neighbor percolation models in two dimensions with a variable interaction range, and include the mean-field limit. We employ a Monte Carlo algorithm whose efficiency depends only weakly on the number of equivalent neighbors within the range of the interactions. We consider 2 classes of models, one in which the interacting neighbors fill a circle, and one in which they fill a square. We first determine the critical points, and then determine the critical exponents. These are, for all finite ranges investigated, in agreement with the exactly known exponents of the nearest-neighbor model.
We perform large-scale simulations of the two-dimensional long-range bond percolation model with algebraically decaying percolation probabilities
∼1/r2+σ, using both conventional ensemble and event-based ensemble methods for system sizes up to
L=16384. We accurately determine the critical points, the universal values of several dimensionless quantities, and the corresponding critical exponents. Our results provide compelling evidence that the system undergoes a crossover from short-range to long-range universality at
σ=2, in contradiction to Sak's criterion. Notably, we observe a pronounced jump in the universal values and critical exponents at
σ=2, a feature absent from previous studies.
We study N-cluster correlation functions in four- and five-dimensional (4D and 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for percolation [Phys. Rev. E 72, 016126 (2005)] using the disjoint-set data structure, and simulate a cylindrical geometry Ld−1 × ∞, with the linear size up to L = 512 for 4D and 128 for 5D. We determine with a high precision all possible N-cluster exponents, for N = 2 and 3, and the universal amplitude for a logarithmic correlation function. From the symmetric correlator with N=2, we obtain the correlation-length critical exponent as 1/ν=1.4610(12) for 4D and 1/ν=1.737(2) for 5D, significantly improving over the existing results. Estimates for the other exponents and the universal logarithmic amplitude have not been reported before to our knowledge. Our work demonstrates the validity of logarithmic conformal field theory and adds to the growing knowledge for high-dimensional percolation.