arXiv:2512.02948

Universality Diagram of Phase Transitions in Long-range Statistical Systems

Tianning Xiao, Zhijie Fan, Youjin Deng

Published Dec 2, 2025• 0 citations

Abstract

The percolation, Ising, and O($n$) models constitute fundamental systems in statistical and condensed matter physics. For short-range-interacting cases, the nature of their phase transitions is well established by renormalization-group theory. However, the universality of the transitions in these models remains elusive when algebraically decaying long-range interactions $\sim 1/r^{d+σ}$ are introduced, where $d$ is the dimensionality and $σ$ is the decay exponent. Building upon insights from Lévy flight, i.e., long-range simple random walk, we propose three universality diagrams in the $(d,σ)$ plane for the percolation model, the O($n$) model, and the Fortuin-Kasteleyn Ising model, respectively. The conjectured universality diagrams are consistent with recent high-precision numerical studies and rigorous mathematical results, offering a unified perspective on critical phenomena in systems with long-range interactions.

DOI: 10.48550/arxiv.2512.02948

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arXiv:2512.02948

Universality Diagram of Phase Transitions in Long-range Statistical Systems

Tianning Xiao, Zhijie Fan, Youjin Deng

Published Dec 2, 2025• 0 citations

arXiv HTML PDF

Abstract

The percolation, Ising, and O(

n

) models constitute fundamental systems in statistical and condensed matter physics. For short-range-interacting cases, the nature of their phase transitions is well established by renormalization-group theory. However, the universality of the transitions in these models remains elusive when algebraically decaying long-range interactions

\sim 1/r^{d+σ}

are introduced, where

d

is the dimensionality and

σ

is the decay exponent. Building upon insights from Lévy flight, i.e., long-range simple random walk, we propose three universality diagrams in the

(d,σ)

plane for the percolation model, the O(

n

) model, and the Fortuin-Kasteleyn Ising model, respectively. The conjectured universality diagrams are consistent with recent high-precision numerical studies and rigorous mathematical results, offering a unified perspective on critical phenomena in systems with long-range interactions.

DOI: 10.48550/arxiv.2512.02948

Related papers

On Sak's criterion for statistical models with long-range interaction

Xiao, Tianning, Liu, Ziyu, Fan, Zhijie +1 more

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, $σ_* = 2 - η_{\mathrm{SR}}$ , 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 $1/r^{2+σ}$ . 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 $R_p$ , the Binder ratio $Q_m$ , 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 $σ_* = 2$ .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, $σ_* = 2 - η_{\mathrm{SR}}$, 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 $1/r^{2+σ}$. 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 $R_p$, the Binder ratio $Q_m$, 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 $σ_* = 2$.

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Jiang Zhou, Ziyu Liu, Pengcheng Hou +2 more

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 $z_c$ , the universal Binder ratio $Q_N^c$ , 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 $σ\le 1$ , 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 $1/2 < σ\leq 1$ , $η$ agrees with the long-range Gaussian-fixed-point prediction $η_{\mathrm{GFP}}=2-σ$ , whereas $Q_N^c$ 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 $n\to0$ limit.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 $z_c$, the universal Binder ratio $Q_N^c$, 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 $σ\le 1$, 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 $1/2 < σ\leq 1$, $η$ agrees with the long-range Gaussian-fixed-point prediction $η_{\mathrm{GFP}}=2-σ$, whereas $Q_N^c$ 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 $n\to0$ limit.

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The percolation, Ising, and O(

n

\sim 1/r^{d+σ}

are introduced, where

d

is the dimensionality and

σ

is the decay exponent. Building upon insights from Lévy flight, i.e., long-range simple random walk, we propose three universality diagrams in the

(d,σ)

plane for the percolation model, the O(

n

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 - η_{\mathrm{SR}}

, 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/r^{2+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

R_p

, the Binder ratio

Q_m

, 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 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.

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 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

z_c

, the universal Binder ratio

Q_N^c

, 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\le 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\leq 1

η

agrees with the long-range Gaussian-fixed-point prediction

η_{\mathrm{GFP}}=2-σ

, whereas

Q_N^c

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\to0

limit.