We perform large-scale simulations of the two-dimensional long-range bond percolation model with algebraically decaying percolation probabilities $\sim 1/r^{2+σ}$, 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 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 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.
Determining the threshold value σ∗ that separates the short-range (SR) and long-range (LR) universality classes in phase transitions remains a controversial issue. While Sak's criterion, σ∗=2−ηSR, has been widely accepted, recent studies of two-dimensional (2D) models with long-range interactions have challenged it. In this work, we focus on the crossover between LR and SR criticality in several classical 2D statistical models, including the XY, Heisenberg, percolation, and Ising models, whose interactions decay as 1/r2+σ. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at σ∗=2. Here, we complete the picture by performing large-scale Monte Carlo simulations of the 2D LR-Ising model, reaching lattice sizes up to L=8192. By analyzing the Fortuin-Kasteleyn critical polynomial Rp, the Binder ratio Qm, and the anomalous dimension η, we obtain convergent and self-consistent evidence that the universality class already changes sharply at σ=2. Taken together, these results establish a unified scenario for LR interacting systems: across all studied models, the crossover from LR to SR universality occurs at σ∗=2.
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.
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.
The two-dimensional (2D) XY model plays a crucial role in statistical and condensed-matter physics. With the introduction of long-range interactions, the system exhibits a richer set of physical phenomena and a crossover between nonclassical and short-range universality classes. In this work, we investigate the 2D XY model with algebraically decaying interactions $\ensuremath{\sim}1/{r}^{2+\ensuremath{\sigma}}$ and provide a comprehensive numerical analysis of its thermodynamic properties. We demonstrate that for $\ensuremath{\sigma}\ensuremath{\le}2$, the system undergoes a second-order phase transition into a ferromagnetic phase characterized by the emergence of long-range order. In the low-temperature phase, due to the presence of the Goldstone mode, the correlation function saturates to a nonzero constant in the form of a power law for $\ensuremath{\sigma}<2$, with decaying exponent $2\ensuremath{-}\ensuremath{\sigma}$, and in the form of the inverse logarithm of distance for $\ensuremath{\sigma}=2$. Moreover, the critical points and exponents are also determined for various $\ensuremath{\sigma}$. We provide compelling evidence that the crossover between nonclassical and short-range regimes occurs at $\ensuremath{\sigma}=2$. This work presents a detailed account of the simulation methodology, extensive numerical data, and new insights into the physics of long-range interacting systems.
We study the one-dimensional long-range self-avoiding walk in the grand-canonical ensemble where the statistical weight of a jump of length r decays algebraically as r−(d+σ). Using large-scale Monte Carlo simulations with an efficient irreversible update scheme, we obtain high-precision estimates of the critical fugacity zc, the universal Binder ratio QNc, the correlation-length exponent ν, and the anomalous dimension η. For σ>1, the critical fugacity varies smoothly with σ, while the Binder ratio and the critical exponents remain consistent with the short-range universality class. For σ≤1, by contrast, the results clearly depart from short-range behavior, identifying σ=1 as the boundary between long-range and short-range regimes. In the long-range Wilson-Fisher regime with 1/2<σ≤1, η agrees with the long-range Gaussian-fixed-point prediction ηGFP=2−σ, whereas QNc and ν vary nontrivially with σ and exhibit discontinuous jumps at σ=1. These findings are in good agreement with the recently proposed universality diagram in the (d,σ) plane for long-range O(n) models, with the self-avoiding walk corresponding to the n→0 limit.