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

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.

Logarithmic finite-size scaling of the self-avoiding walk at four dimensions.

The n-vector spin model, which includes the self-avoiding walk (SAW) as a special case for the n→0 limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate the SAW on 4D hypercubic lattices with periodic boundary conditions by an irreversible Berretti-Sokal algorithm up to linear size L=768. From an unwrapped end-to-end distance, we obtain the critical fugacity as z_{c}=0.147622380(2), improving over the existing result z_{c}=0.1476223(1) by 50 times. Such a precisely estimated critical point enables us to perform a systematic study of the finite-size scaling of 4D SAW for various quantities. Our data indicate that near z_{c}, the scaling behavior of the free energy simultaneously contains a scaling term from the Gaussian fixed point and the other accounting for multiplicative logarithmic corrections. In particular, it is clearly observed that the critical magnetic susceptibility and the specific heat logarithmically diverge as χ∼L^{2}(lnL)^{2y[over ̂]_{h}} and C∼(lnL)^{2y[over ̂]_{t}}, and the logarithmic exponents are determined as y[over ̂]_{h}=0.251(2) and y[over ̂]_{t}=0.25(3), in excellent agreement with the field theoretical prediction y[over ̂]_{h}=y[over ̂]_{t}=1/4. Our results provide a strong support for the recently conjectured finite-size scaling form for the O(n) universality classes at 4D.

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.

Random-Length Random Walks and Finite-Size Scaling in High Dimensions.

We address a long-standing debate regarding the finite-size scaling (FSS) of the Ising model in high dimensions, by introducing a random-length random walk model, which we then study rigorously. We prove that this model exhibits the same universal FSS behavior previously conjectured for the self-avoiding walk and Ising model on finite boxes in high-dimensional lattices. Our results show that the mean walk length of the random walk model controls the scaling behavior of the corresponding Green's function. We numerically demonstrate the universality of our rigorous findings by extensive Monte Carlo simulations of the Ising model and self-avoiding walk on five-dimensional hypercubic lattices with free and periodic boundaries.

Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions.

We give an intuitive geometric explanation for the apparent breakdown of standard finite-size scaling in systems with periodic boundaries above the upper critical dimension. The Ising model and self-avoiding walk are simulated on five-dimensional hypercubic lattices with free and periodic boundary conditions, by using geometric representations and recently introduced Markov-chain Monte Carlo algorithms. We show that previously observed anomalous behavior for correlation functions, measured on the standard Euclidean scale, can be removed by defining correlation functions on a scale which correctly accounts for windings.

Leaf-excluded percolation in two and three dimensions.

We introduce the leaf-excluded percolation model, which corresponds to independent bond percolation conditioned on the absence of leaves (vertices of degree one). We study the leaf-excluded model on the square and simple-cubic lattices via Monte Carlo simulation, using a worm-like algorithm. By studying wrapping probabilities, we precisely estimate the critical thresholds to be 0.3552475(8) (square) and 0.185022(3) (simple-cubic). Our estimates for the thermal and magnetic exponents are consistent with those for percolation, implying that the phase transition of the leaf-excluded model belongs to the standard percolation universality class.

Geometric structure of percolation clusters.

We investigate the geometric properties of percolation clusters by studying square-lattice bond percolation on the torus. We show that the density of bridges and nonbridges both tend to 1/4 for large system sizes. Using Monte Carlo simulations, we study the probability that a given edge is not a bridge but has both its loop arcs in the same loop and find that it is governed by the two-arm exponent. We then classify bridges into two types: branches and junctions. A bridge is a branch iff at least one of the two clusters produced by its deletion is a tree. Starting from a percolation configuration and deleting the branches results in a leaf-free configuration, whereas, deleting all bridges produces a bridge-free configuration. Although branches account for ≈43% of all occupied bonds, we find that the fractal dimensions of the cluster size and hull length of leaf-free configurations are consistent with those for standard percolation configurations. By contrast, we find that the fractal dimensions of the cluster size and hull length of bridge-free configurations are given by the backbone and external perimeter dimensions, respectively. We estimate the backbone fractal dimension to be 1.643 36(10).

High-precision Monte Carlo study of directed percolation in (d+1) dimensions.

We present a Monte Carlo study of the bond- and site-directed (oriented) percolation models in (d+1) dimensions on simple-cubic and body-centered-cubic lattices, with 2 ≤ d ≤ 7. A dimensionless ratio is defined, and an analysis of its finite-size scaling produces improved estimates of percolation thresholds. We also report improved estimates for the standard critical exponents. In addition, we study the probability distributions of the number of wet sites and radius of gyration, for 1 ≤ d ≤ 7.

Bond and site percolation in three dimensions.

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512, and estimate the percolation thresholds to be p(c)(bond)=0.24881182(10) and p(c)(site)=0.3116077(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents 1/ν=1.1410(15), ß/ν=0.47705(15), the leading correction exponent y(i)=-1.2(2), and the shortest-path exponent d(min)=1.3756(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent ≈-2, rather than y(i)≈-1.2.

Crossover from isotropic to directed percolation.

We generalize the directed percolation (DP) model by relaxing the strict directionality of DP such that propagation can occur in either direction but with anisotropic probabilities. We denote the probabilities as p(↓) = pp(d) and p(↑) = p(1-p(d)), with p representing the average occupation probability and p(d) controlling the anisotropy. The Leath-Alexandrowicz method is used to grow a cluster from an active seed site. We call this model with two main growth directions biased directed percolation (BDP). Standard isotropic percolation (IP) and DP are the two limiting cases of the BDP model, corresponding to p(d) =1/2 and p(d) = 0,1 respectively. In this work, besides IP and DP, we also consider the 1/2 < p(d) <1 region. Extensive Monte Carlo simulations are carried out on the square and the simple-cubic lattices, and the numerical data are analyzed by finite-size scaling. We locate the percolation thresholds of the BDP model for p(d) = 0.6 and 0.8, and determine various critical exponents. These exponents are found to be consistent with those for standard DP. We also determine the renormalization exponent associated with the asymmetric perturbation due to p(d)-1/2 ≠ 0 near IP, and confirm that such an asymmetric scaling field is relevant at IP.

Shortest-path fractal dimension for percolation in two and three dimensions.

We carry out a high-precision Monte Carlo study of the shortest-path fractal dimension d(min) for percolation in two and three dimensions, using the Leath-Alexandrowicz method which grows a cluster from an active seed site. A variety of quantities are sampled as a function of the chemical distance, including the number of activated sites, a measure of the radius, and the survival probability. By finite-size scaling, we determine d(min)=1.13077(2) and 1.3756(6) in two and three dimensions, respectively. The result in two dimensions rules out the recently conjectured value d(min)=217/192 [Deng et al., Phys. Rev. E 81, 020102(R) (2010)].

Autocorrelations in the totally asymmetric simple exclusion process and Nagel-Schreckenberg model.

We study via Monte Carlo simulation the dynamics of the Nagel-Schreckenberg model on a finite system of length L with open boundary conditions and parallel updates. We find numerically that in both the high and low density regimes the autocorrelation function of the system density behaves like 1-|t|/τ with a finite support [-τ,τ] . This is in contrast to the usual exponential decay typical of equilibrium systems. Furthermore, our results suggest that in fact τ=L/c , and in the special case of maximum velocity v{max}=1 (corresponding to the totally asymmetric simple exclusion process) we can identify the exact dependence of c on the input, output and hopping rates. We also emphasize that the parameter τ corresponds to the integrated autocorrelation time, which plays a fundamental role in quantifying the statistical errors in Monte Carlo simulations of these models.