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arXiv:2608.20750

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

Ziyu Liu, Tianning Xiao, Zhijie Fan, Youjin Deng

Published Aug 21, 2026• 0 citations
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Abstract

We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+σ}$, establishing the universality diagram in the long-range (LR) regime for $σ\le2$. 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 ($1<σ\le2$), LR Wilson--Fisher B ($2/3<σ\le1$), and LR mean-field (MF) ($0<σ\le2/3$). 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 $σ\simeq3/2$ 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.

DOI
10.48550/arxiv.2608.20750

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We perform large-scale simulations of the two-dimensional long-range bond percolation model with algebraically decaying percolation probabilities ∼1/r2+σ\sim 1/r^{2+σ}∼1/r2+σ, using both conventional ensemble and event-based ensemble methods for system sizes up to L=16384L=16384L=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=2σ= 2σ=2, in contradiction to Sak's criterion. Notably, we observe a pronounced jump in the universal values and critical exponents at σ=2σ= 2σ=2, a feature absent from previous studies.
We study the one-dimensional long-range self-avoiding walk in the grand-canonical ensemble where the statistical weight of a jump of length rrr decays algebraically as r−(d+σ)r^{-(d+σ)}r−(d+σ). Using large-scale Monte Carlo simulations with an efficient irreversible update scheme, we obtain high-precision estimates of the critical fugacity zcz_czc​, the universal Binder ratio QNcQ_N^cQNc​, the correlation-length exponent ννν, and the anomalous dimension ηηη. For σ>1σ>1σ>1, the critical fugacity varies smoothly with σσσ, while the Binder ratio and the critical exponents remain consistent with the short-range universality class. For σ≤1σ\le 1σ≤1, by contrast, the results clearly depart from short-range behavior, identifying σ=1σ=1σ=1 as the boundary between long-range and short-range regimes. In the long-range Wilson-Fisher regime with 1/2<σ≤11/2 < σ\leq 11/2<σ≤1, ηηη agrees with the long-range Gaussian-fixed-point prediction ηGFP=2−ση_{\mathrm{GFP}}=2-σηGFP​=2−σ, whereas QNcQ_N^cQNc​ and ννν vary nontrivially with σσσ and exhibit discontinuous jumps at σ=1σ=1σ=1. These findings are in good agreement with the recently proposed universality diagram in the (d,σ)(d,σ)(d,σ) plane for long-range O(n)(n)(n) models, with the self-avoiding walk corresponding to the n→0n\to0n→0 limit.
Determining the threshold value σ∗σ_*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−ηSRσ_* = 2 - η_{\mathrm{SR}}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+σ1/r^{2+σ}1/r2+σ. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at σ∗=2σ_* = 2σ∗​=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=8192L = 8192L=8192. By analyzing the Fortuin-Kasteleyn critical polynomial RpR_pRp​, the Binder ratio QmQ_mQm​, and the anomalous dimension ηηη, we obtain convergent and self-consistent evidence that the universality class already changes sharply at σ=2σ= 2σ=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∗=2σ_* = 2σ∗​=2.
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.
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