Critical percolation clusters in seven dimensions and on a complete graph

Hybrid universality classes of systemic cascades

Cascades are self-reinforcing processes underlying the systemic risk of many complex systems. Understanding the universal aspects of these phenomena is of fundamental interest, yet typically bound to numerical observations in ad-hoc models and limited insights. Here, we develop a unifying approach that reveals two distinct universality classes of cascades determined by the global symmetry of the cascading process. We provide hyperscaling arguments predicting hybrid critical phenomena characterized by a combination of both mean-field spinodal exponents and d-dimensional corrections, and show how parity invariance influences the geometry and lifetime of critical avalanches. Our theory applies to a wide range of networked systems in arbitrary dimensions, as we demonstrate by simulations encompassing classic and novel cascade models, revealing universal principles of cascade critical phenomena amenable to experimental validation.

Interplay of the complete-graph and Gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension

For statistical mechanical systems with continuous phase transitions, there are two closely related but subtly different mean-field treatments, the Gaussian fixed point (GFP) in the renormalization group framework and the Landau mean-field theory or the complete-graph (CG) asymptotics. By large-scale Monte Carlo simulations, we systematically study the interplay of the GFP and CG effects to the finite-size scaling of percolation above the upper critical dimension d_{c}=6 with periodic and cylindrical boundary conditions. Our results suggest that, with periodic boundaries, the unwrapped correlation length scales as L^{d/6} at the critical point, diverging faster than L above d_{c}. As a consequence, the scaling behaviors of macroscopic quantities with respect to the linear system size L follow the CG asymptotics. The distance-dependent properties, such as the short-distance behavior of the two-point correlation function and the Fourier transformed quantities with nonzero modes, are still controlled by the GFP. With cylindrical boundaries, due to the interplay of the GFP and CG effects, the correlation length along the axial direction of the cylinder scales as ξ_{L}∼L^{(d-1)/5} within the critical window of size O(L^{-2(d-1)/5}), distinct from periodic boundary. A field-theoretical calculation for deriving the scaling of ξ_{L} is also presented. Moreover, the one-point surface correlation function along the axial direction of the cylinder is observed to scale as τ^{(1-d)/2} when the distance τ is short, but then enter a plateau of order L^{-3(d-1)/5} before it decays significantly fast.

Emergence of biconnected clusters in explosive percolation

By introducing a simple competition mechanism for bond insertion in random graphs, explosive percolation exhibits a sharp phase transition with rich critical phenomena. We investigate high-order connectivity in explosive percolation using an event-based ensemble, focusing on biconnected clusters, where any two sites are connected by at least two independent paths. Our numerical analysis confirms that explosive percolation with different intracluster bond competition rules shares the same percolation threshold and universality, with biconnected clusters percolating simultaneously with simply connected clusters. However, the volume fractal dimension d_{f}^{'} of biconnected clusters varies depending on the competition rules of intracluster bonds. The size distribution of biconnected clusters exhibits double-scaling behavior: large clusters follow the standard Fisher exponent derived from the hyperscaling relation τ^{'}=1+1/d_{f}^{'}, while small clusters display a modified Fisher exponent τ_{0}<τ^{'}. These findings provide insights into the intricate nature of connectivity in explosive percolation.

Energy Formulation for Infinite Structures: Order Parameter for Percolation, Critical Bonds, and Power-Law Scaling of Contact-Based Transport

Investigating heterogeneous materials' microstructure, often simulated using periodic images, is crucial for understanding their physical traits. We propose a generic spring-based representation for periodic two-component structures. The equilibrium energy in this framework serves as an order parameter, offering an analytical expression for wrapping and introducing the concept of critical bonds. We show that these minimum bonds for depercolation can be efficiently detected. The number of critical bonds scales with system size, accurately capturing contact-based transport's scaling. This approach holds potential to analyze functional robustness of networks.

Finite-size scaling of the high-dimensional Ising model in the loop representation

Besides its original spin representation, the Ising model is known to have the Fortuin-Kasteleyn (FK) bond and loop representations, of which the former was recently shown to exhibit two upper critical dimensions (d_{c}=4,d_{p}=6). Using a lifted worm algorithm, we determine the critical coupling as K_{c}=0.07770891(4) for d=7, which significantly improves over the previous results, and then study critical geometric properties of the loop Ising clusters on tori for spatial dimensions d=5 to 7. We show that as the spin representation, the loop Ising model has only one upper critical dimension at d_{c}=4. However, sophisticated finite-size scaling (FSS) behaviors, such as two length scales, two configuration sectors, and two scaling windows, still exist as the interplay effect of the Gaussian fixed point and complete-graph asymptotics. Moreover, using the loop-cluster algorithm, we provide an intuitive understanding of the emergence of the percolation-like upper critical dimension d_{p}=6 in the FK-Ising model. As a consequence, a unified physical picture is established for the FSS behaviors in all three representations of the Ising model above d_{c}=4.

Geometric properties of the complete-graph Ising model in the loop representation

The exact solution of the Ising model on the complete graph (CG) provides an important, though mean-field, insight for the theory of continuous phase transitions. Besides the original spin, the Ising model can be formulated in the Fortuin-Kasteleyn random cluster and the loop representation, in which many geometric quantities have no correspondence in the spin representations. Using a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop representation and, based on theoretical and numerical analyses, obtain a number of exact results including volume fractal dimensions and scaling forms. Moreover, by combining with the loop-cluster algorithm, we demonstrate how the loop representation can provide an intuitive understanding to the recently observed rich geometric phenomena in the random-cluster representation, including the emergence of two configuration sectors, two length scales, and two scaling windows.

Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions

Recently, we argued [Chin. Phys. Lett. 39, 080502 (2022)0256-307X10.1088/0256-307X/39/8/080502] that the Ising model simultaneously exhibits two upper critical dimensions (d_{c}=4,d_{p}=6) in the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we perform a systematic study of the FK Ising model on hypercubic lattices with spatial dimensions d from 5 to 7, and on the complete graph. We provide a detailed data analysis of the critical behaviors of a variety of quantities at and near the critical points. Our results clearly show that many quantities exhibit distinct critical phenomena for 4<d<6 and d≥6, and thus strongly support the argument that 6 is also an upper critical dimension. Moreover, for each studied dimension, we observe the existence of two configuration sectors, two lengthscales, as well as two scaling windows, and thus two sets of critical exponents are needed to describe these behaviors. Our finding enriches the understanding of the critical phenomena in the Ising model.

Backbone and shortest-path exponents of the two-dimensional Q-state Potts model

We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional Q-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents d_{B}=1.7320(3) and 1.794(2) for Q=2,3, respectively. However, for large Q, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O(n) loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with Q=n^{2}. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops d_{B}=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6) for Q=1,2,3,2+sqrt[3],4, respectively, and d_{min}=1.0945(2),1.0675(3),1.0475(3),1.0322(4) for Q=2,3,2+sqrt[3],4, respectively. Our estimates significantly improve over the existing results for both d_{B} and d_{min}. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of n for the leading correction exponent.

Percolation thresholds on high-dimensional D_{n} and E_{8}-related lattices

The site and bond percolation problems are conventionally studied on (hyper)cubic lattices, which afford straightforward numerical treatments. The recent implementation of efficient simulation algorithms for high-dimensional systems now also facilitates the study of D_{n} root lattices in n dimensions as well as E_{8}-related lattices. Here, we consider the percolation problem on D_{n} for n=3 to 13 and on E_{8} relatives for n=6 to 9. Precise estimates for both site and bond percolation thresholds obtained from invasion percolation simulations are compared with dimensional series expansion based on lattice animal enumeration for D_{n} lattices. As expected, the bond percolation threshold rapidly approaches the Bethe lattice limit as n increases for these high-connectivity lattices. Corrections, however, exhibit clear yet unexplained trends. Interestingly, the finite-size scaling exponent for invasion percolation is found to be lattice and percolation-type specific.

Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph

The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.

Complete graph and Gaussian fixed-point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries

We present an extensive Markov chain Monte Carlo study of the finite-size scaling behavior of the Fortuin-Kasteleyn Ising model on five-dimensional hypercubic lattices with periodic boundary conditions. We observe that physical quantities, which include the contribution of the largest cluster, exhibit complete graph asymptotics. However, for quantities where the contribution of the largest cluster is removed, we observe that the scaling behavior is mainly controlled by the Gaussian fixed point. Our results therefore suggest that both scaling predictions, i.e., the complete graph and the Gaussian fixed point asymptotics, are needed to provide a complete description for the five-dimensional finite-size scaling behavior on the torus.

Finite-size scaling of O(n) systems at the upper critical dimensionality

Logarithmic finite-size scaling of the O(n) universality class at the upper critical dimensionality (d _c = 4) has a fundamental role in statistical and condensed-matter physics and important applications in various experimental systems. Here, we address this long-standing problem in the context of the n-vector model (n = 1, 2, 3) on periodic four-dimensional hypercubic lattices. We establish an explicit scaling form for the free-energy density, which simultaneously consists of a scaling term for the Gaussian fixed point and another term with multiplicative logarithmic corrections. In particular, we conjecture that the critical two-point correlation g(r, L), with L the linear size, exhibits a two-length behavior: follows [Formula: see text] governed by the Gaussian fixed point at shorter distances and enters a plateau at larger distances whose height decays as [Formula: see text] with [Formula: see text] a logarithmic correction exponent. Using extensive Monte Carlo simulations, we provide complementary evidence for the predictions through the finite-size scaling of observables, including the two-point correlation, the magnetic fluctuations at zero and nonzero Fourier modes and the Binder cumulant. Our work sheds light on the formulation of logarithmic finite-size scaling and has practical applications in experimental systems.

Bond percolation between k separated points on a square lattice

We consider a percolation process in which k points separated by a distance proportional the system size L simultaneously connect together (k>1), or a single point at the center of a system connects to the boundary (k=1), through adjacent connected points of a single cluster. These processes yield new thresholds p[over ¯]_{ck} defined as the average value of p at which the desired connections first occur. These thresholds not sharp, as the distribution of values of p_{ck} for individual samples remains broad in the limit of L→∞. We study p[over ¯]_{ck} for bond percolation on the square lattice and find that p[over ¯]_{ck} are above the normal percolation threshold p_{c}=1/2 and represent specific supercritical states. The p[over ¯]_{ck} can be related to integrals over powers of the function P_{∞}(p) equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of P_{∞}(p) on L×L systems that for L→∞, p[over ¯]_{c1}=0.51755(5), p[over ¯]_{c2}=0.53219(5), p[over ¯]_{c3}=0.54456(5), and p[over ¯]_{c4}=0.55527(5). The percolation thresholds p[over ¯]_{ck} remain the same, even when the k points are randomly selected within the lattice. We show that the finite-size corrections scale as L^{-1/ν_{k}} where ν_{k}=ν/(kβ+1), with β=5/36 and ν=4/3 being the ordinary percolation critical exponents, so that ν_{1}=48/41, ν_{2}=24/23, ν_{3}=16/17, ν_{4}=6/7, etc. We also study three-point correlations in the system and show how for p>p_{c}, the correlation ratio goes to 1 (no net correlation) as L→∞, while at p_{c} it reaches the known value of 1.022.

Geometric properties of the Fortuin-Kasteleyn representation of the Ising model

We present a Monte Carlo study of the geometric properties of Fortuin-Kasteleyn (FK) clusters of the Ising model on square [two-dimensional (2D)] and simple-cubic [three-dimensional (3D)] lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio and yields a high-precision critical coupling as K_{c}(3D)=0.221654631(8). We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size C_{1} follows a single-variable function as P(C_{1},L)dC_{1}=P[over ̃](x)dx with x≡C_{1}/L^{d_{F}} (L is the linear size), where the fractal dimension d_{F} is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution P[over ̃](x) in three dimensions, and attributed to the different approaching behaviors for K→K_{c}+0^{±}. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as d_{min}(3D)=1.25936(12) and d_{min}(2D)=1.0940(2). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as d_{B}(3D)=2.1673(15) and d_{B}(2D)=1.7321(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents d_{min} and d_{B} either improve over the existing results or have not been reported yet. We believe that these numerical results would provide a testing ground in the development of further theoretical treatments of the 3D Ising model.

Percolation thresholds and Fisher exponents in hypercubic lattices

We use invasion percolation to compute highly accurate numerical values for bond and site percolation thresholds p_{c} on the hypercubic lattice Z^{d} for d=4,...,13. We also compute the Fisher exponent τ governing the cluster size distribution at criticality. Our results support the claim that the mean-field value τ=5/2 holds for d≥6, with logarithmic corrections to power-law scaling at d=6.