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

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 Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, perform a systematic study of FK on hypercubic lattices with spatial d from 5 to 7, and complete graph. We provide detailed data analysis behaviors variety quantities at near points. Our results clearly show many exhibit distinct...

We carry out a high-precision Monte Carlo study of the shortest-path fractal dimension $\dm$ for percolation in two and three dimensions, using Leath-Alexandrowicz method which grows cluster from an active seed site. A variety quantities are sampled as function chemical distance, including number activated sites, measure radius, survival probability. By finite-size scaling, we determine $\dm = 1.130 77(2)$ $1.375 6(6)$ respectively. The result 2D rules recently conjectured value...

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

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

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

We investigate the geometric properties of percolation clusters by studying square-lattice bond on torus. show that density bridges and nonbridges both tend to 1/4 for large system sizes. Using Monte Carlo simulations, we study probability a given edge is not bridge but has its loop arcs in same find it governed two-arm exponent. then classify into two types: branches junctions. A branch iff at least one produced deletion tree. Starting from configuration deleting results leaf-free...

Besides its original spin representation, the Ising model is known to have Fortuin-Kasteleyn (FK) bond and loop representations, of which former was recently shown exhibit two upper critical dimensions $({d}_{c}=4,{d}_{p}=6)$. Using a lifted worm algorithm, we determine coupling as ${K}_{c}=0.077\phantom{\rule{0.16em}{0ex}}708\phantom{\rule{0.16em}{0ex}}91(4)$ for $d=7$, significantly improves over previous results, then study geometric properties clusters on tori spatial $d=5$ 7. We show...

The upper critical dimension of the Ising model is known to be d c = 4, above which behavior regarded trivial. We hereby argue from extensive simulations that, in random-cluster representation, simultaneously exhibits two dimensions at ( p 6), and clusters for ≥ , except largest one, are governed by exponents percolation universality. predict a rich variety geometric properties then provide strong evidence 4 7 on complete graphs. Our findings significantly advance understanding model,...

We generalize the directed percolation (DP) model by relaxing strict directionality of DP such that propagation can occur in either direction but with anisotropic probabilities. denote probabilities as $p_{\downarrow}= p \cdot p_d$ and $p_{\uparrow}=p (1-p_d)$, $p $ representing average occupation probability $p_d$ controlling anisotropy. The Leath-Alexandrowicz method is used to grow a cluster from an active seed site. call this two main growth directions {\em biased percolation} (BDP)....

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

We study the variable-length ensemble of self-avoiding walks on complete graph. obtain leading order asymptotics mean and variance walk length, as number vertices goes to infinity. Central limit theorems for length are also established, in various regimes fugacity. Particular attention is given sequences fugacities that converge critical point, effect rate convergence these fugacity limiting studied detail. Physically, this corresponds studying asymptotic a general class pseudocritical points.

Abstract We study the two-point functions of a general class random-length random walks (RLRWs) on finite boxes in <?CDATA $\mathbb{Z}^d$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> with $d\unicode{x2A7E}3$?> <mml:mtext>⩾</mml:mtext> <mml:mn>3</mml:mn> , and provide precise asymptotics for their behaviour. show that box side length L...

In this article, a worm algorithm is used to study the problem of Ising model on complete graph. The approach shows connections between different representations and leads better understanding its scaling properties critical behavior.

The n-vector spin model, which includes the self-avoiding walk (SAW) as a special case for n→0 limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate SAW on 4D hypercubic lattices with periodic boundary conditions by irreversible Berretti-Sokal algorithm up to linear size L=768. From unwrapped end-to-end distance, we obtain fugacity z_{c}=0.147622380(2), improving over existing result z_{c}=0.1476223(1) 50 times. Such precisely estimated point enables us...

The Fortuin-Kasteleyn (FK) random cluster model, which can be exactly mapped from the $q$-state Potts spin is a correlated bond percolation model. By extensive Monte Carlo simulations, we study FK representation of critical Ising model ($q=2$) on finite complete graph, i.e. mean-field We provide strong numerical evidence that configuration space for $q=2$ contains an asymptotically vanishing sector in quantities exhibit same finite-size scaling as uncorrelated ($q=1$) graph. Moreover,...

We study unwrapped two-point functions for the Ising model, self-avoiding walk and a random-length loop-erased random on high-dimensional lattices with periodic boundary conditions. While standard of these models have been observed to display an anomalous plateau behaviour, are shown mean-field behaviour. Moreover, we argue that asymptotic behaviour torus can be understood in terms function model Zd. A precise description is derived latter. Finally, consider natural notion length, show...

We study the two-point functions of a general class random-length random walks on finite boxes in $\ZZ^d$ with $d\ge3$, and provide precise asymptotics for their behaviour. show that finite-box function is asymptotic to infinite-lattice when typical walk length $o(L^2)$, but develops plateau $\Omega(L^2)$. also numerically moments limiting distributions self-avoiding Ising model five-dimensional tori, find they agree asymptotically known results complete graph, both at critical point broad...

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

Field-theoretical calculations predict that, at the upper critical dimension $d_c=4$, finite-size scaling (FSS) behaviors of Ising model would be modified by multiplicative logarithmic corrections with thermal and magnetic correction exponents $(\hat{y}_t, \hat{y}_h)=(1/6,1/4)$. Using high-efficient cluster algorithms lifted worm algorithm, we present a systematic study FSS four-dimensional in Fortuin-Kasteleyn (FK) bond loop representations. Our numerical results reveal various geometric...