The two-dimensional Potts model can be studied either in terms of the original Q-component spins, or in the geometrical reformulation via Fortuin-Kasteleyn (FK) clusters. While the FK representation makes sense for arbitrary real values of Q by construction, it was only shown very recently that the spin representation can be promoted to the same level of generality. In this paper we show how to define the Potts model in terms of observables that simultaneously keep track of the spin and FK degrees of freedom. This is first done algebraically in terms of a transfer matrix that couples three different representations of a partition algebra. Using this, one can study correlation functions involving any given number of propagating spin clusters with prescribed colours, each of which contains any given number of distinct FK clusters. For 0 <= Q <= 4 the corresponding critical exponents are all of the Kac form h_{r,s}, with integer indices r,s that we determine exactly both in the bulk and in the boundary versions of the problem. In particular, we find that the set of points where an FK cluster touches the hull of its surrounding spin cluster has fractal dimension d_{2,1} = 2 - 2 h_{2,1}. If one constrains this set to points where the neighbouring spin cluster extends to infinity, we show that the dimension becomes d_{1,3} = 2 - 2 h_{1,3}. Our results are supported by extensive transfer matrix and Monte Carlo computations.
We investigate the dependence of the critical Binder cumulant of the magnetization and the largest Fortuin-Kasteleyn cluster on the boundary conditions and aspect ratio of the underlying square Ising lattices. By means of the Swendsen-Wang algorithm, we generate numerical data for large system sizes and we perform a detailed finite-size scaling analysis for several values of the aspect ratio $r$, for both periodic and free boundary conditions. We estimate the universal probability density functions of the largest Fortuin-Kasteleyn cluster and we compare it to those of the magnetization at crit…
It was pointed out by de Arcangelis et al. [Europhys. Lett. 14 (1991), 515] that the correct understanding of the percolation phenomenon of the Fortuin-Kasteleyn cluster in the Edwards-Anderson model is important since a dynamical transition, which is characterized by a parameter called the Hamming distance or damage, and the percolation transition are related to a transition for a signal propagating between spins. We show analytically the percolation thresholds of the Fortuin-Kasteleyn cluster for a Potts gauge glass model, which is an extended model of the Edwards-Anderson model, on random g…
We analytically show the percolation thresholds of the Fortuin-Kasteleyn cluster for the Edwards-Anderson Ising model on random graphs with arbitrary degree distributions. The results on the Nishimori line are shown. We obtain the results for the +-J model, the diluted +-J model, and the Gaussian model, by applying an extension of a criterion for the random graphs with arbitrary degree distributions. The results for the infinite-range $\pm J$ model and the Sherrington-Kirkpatrick model are also shown.
The Fortuin-Kasteleyn and heat-bath damage spreading temperatures T_{FK}(p) and T_{ds}(p) are studied on random bond Ising models of dimension two to five and as functions of the ferromagnetic interaction probability p; the conjecture that T_{ds}(p) ~ T_{FK}(p) is tested. It follows from a statement by Nishimori that in any such system exact coordinates can be given for the intersection point between the Fortuin-Kasteleyn T_{FK}(p) transition line and the Nishimori line, [p_{NL,FK},T_{NL,FK}]. There are no finite size corrections for this intersection point. In dimension three, at the intersec…
Topological properties of Fortuin-Kasteleyn clusters are studied on the torus. Namely, the probability that their topology yields a given subgroup of the first homology group of the torus is computed for Q=1, 2, 3 and 4. The expressions generalize those obtained by Pinson for percolation (Q=1). Numerical results are also presented for three tori of different moduli. They agree with the theoretical predictions for Q=1, 2 and 3. For Q=4 agreement is not ruled out but what seems logarithmic corrections makes it harder to decide.
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 \textit{both} scaling predictions, i.e. the complete graph \text…
The variance in the winding number of various random fractal curves, including the self-avoiding walk, the loop-erased random walk, contours of FK clusters, and stochastic Loewner evolution, have been studied by numerous researchers. Usually the focus has been on the winding at the endpoints. We measure the variance in winding number at typical points along the curve. More generally, we study the winding at points where k strands come together, and some adjacent strands may be conditioned not to hit each other. The measured values are consistent with an interesting conjecture.
We derive the exact internal energies and the rigorous upper bounds of specific heats for several spin glass models by applying the gauge theory to the Fortuin-Kasteleyn representation which is a representation based on a percolation picture for spin-spin correlation. The results are derived on the Nishimori lines which are special lines on the phase diagrams. As the spin glass models, the +-J Ising model and a Potts gauge glass model are studied. The present solutions agree with the previous solutions. The derivation of the solutions by the present method must be useful for understanding the …
A Fortuin-Kasteleyn cluster on a torus is said to be of type $\{a,b\}, a,b\in\mathbb Z$, if it possible to draw a curve belonging to the cluster that winds $a$ times around the first cycle of the torus as it winds $-b$ times around the second. Even though the $Q$-Potts models make sense only for $Q$ integers, they can be included into a family of models parametrized by $β=\sqrt{Q}$ for which the Fortuin-Kasteleyn clusters can be defined for any real $β\in (0,2]$. For this family, we study the probability $π({\{a,b\}})$ of a given type of clusters as a function of the torus modular parameter $τ…
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 obser…
We consider the coupling from the past implementation of the random-cluster heat-bath process, and study its random running time, or coupling time. We focus on hypercubic lattices embedded on tori, in dimensions one to three, with cluster fugacity at least one. We make a number of conjectures regarding the asymptotic behaviour of the coupling time, motivated by rigorous results in one dimension and Monte Carlo simulations in dimensions two and three. Amongst our findings, we observe that, for generic parameter values, the distribution of the appropriately standardized coupling time converges t…
The tricritical behavior of the two-dimensional $q$-state Potts model with vacancies for $1\leq q \leq4$ is argued to be encoded in the fractal structure of the geometrical spin clusters of the pure model. The close connection between the critical properties of the pure model and the tricritical properties of the diluted model is shown to be reflected in an intimate relation between Fortuin-Kasteleyn and geometrical clusters: The same transformation mapping the two critical regimes onto each other also maps the two cluster types onto each other. The map conserves the central charge, so that bo…
The conjectured exact percolation thresholds of the Fortuin-Kasteleyn cluster for the +-J Ising spin glass model are theoretically shown based on a conjecture. It is pointed out that the percolation transition of the Fortuin-Kasteleyn cluster for the spin glass model is related to a dynamical transition for the freezing of spins. The present results are obtained as locations of points on the so-called Nishimori line, which is a special line in the phase diagram. We obtain TFK = 2 / ln [z / (z - 2)] and pFK = z / [2 (z - 1)] for the Bethe lattice, TFK -> infinity and pFK -> 1 / 2 for the infini…
We compare the correlation times of the Sweeny and Gliozzi dynamics for two-dimensional Ising and three-state Potts models, and the three-dimensional Ising model for the simulations in the percolation prepresentation. The results are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found that Sweeny and Gliozzi dynamics have essentially the same dynamical critical behavior. Contrary to Gliozzi's claim (cond-mat/0201285), the Gliozzi dynamics has critical slowing down comparable to that of other cluster methods. For the two-dimensional Ising model, both Sweeny and Gliozzi dyna…
Recently, two of us argued that the probability that an FK cluster in the Q-state Potts model connects three given points is related to the time-like Liouville three-point correlation function. Moreover, they predicted that the FK three-point connectivity has a prefactor which unveils the effects of a discrete symmetry, reminiscent of the S_Q permutation symmetry of the Q=2,3,4 Potts model. Their theoretical prediction has been checked for the case of percolation, corresponding to Q=1. We revisit the derivation of the time-like Liouville correlator given by Al. Zamolodchikov and show that this…
We present a Monte Carlo study of the Fortuin-Kasteleyn (FK) clusters of the Ising model on the square (2D) and simple-cubic (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 $K_c(3\rm D)=0.221\,654\,631(8)$. We then study geometric properties of the FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size $C_1$ follows a single-variab…
For the two-dimensional Q-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number N_Gamma of clusters that intersect a given contour Gamma. To leading order, N_Gamma is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of Gamma. These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition we find agreement with these predictions from larg…
We report on numerical investigation of fractal properties of critical interfaces in two-dimensional Potts models. Algorithms for finding percolating interfaces of Fortuin-Kasteleyn clusters, their external perimeters and interfaces of spin clusters are presented. Fractal dimensions are measured and compared to exact theoretical predictions.
We analyse parafermionic operators in the Q-state Potts model from three different perspectives. First, we explicitly construct lattice holomorphic observables in the Fortuin-Kasteleyn representation, and point out some special simplifying features of the particular case Q=2 (Ising model). In particular, away from criticality, we find a lattice generalisation of the massive Majorana fermion equation. We also compare the parafermionic scaling dimensions with known results from CFT and Coulomb gas methods in the continuum. Finally, we show that expectation values of these parafermions correspond…
Renormalization group and Coulomb gas mappings are used to derive theoretical predictions for the corrections to the exactly known asymptotic fractal masses of the hull, external perimeter, singly connected bonds and total mass of the Fortuin-Kasteleyn clusters for two-dimensional $q$-state Potts models at criticality. For q=4 these include exact logarithmic (as well as loglog) corrections.
We provide a representation for the scaling limit of the d=2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d=3 and to Potts models are also considered.
We calculated numerically the fractal dimension of the boundaries of the Fortuin-Kasteleyn clusters of the $q$-state Potts model for integer and non-integer values of $q$ on the square lattice. In addition we calculated with high accuracy the fractal dimension of the boundary points of the same clusters on the square domain. Our calculation confirms that this curves can be described by SLE$_κ$.
We give a new sampling algorithm for the Potts model based on the Fortuin-Kasteleyn transformation. The method produces independent samples and sums up a large number of configurations for each sweep. The partition function and thermodynamic averages for all values of the temperature can be computed from a single run. We compare the results with exact 2D Ising model.
The deletion--contraction algorithm is perhaps the most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin--Kasteleyn in statistical physics. Prior to this work, deletion--contraction was also the fastest known general-purpose algorithm for these invariants, running in time roughly proportional to the number of spanning trees in the input graph. Here, we give a substantially …
We revisit in this paper the problem of connectivity correlations in the Fortuin-Kasteleyn cluster representation of the two-dimensional $Q$-state Potts model conformal field theory. In a recent work [M. Picco, S. Ribault and R. Santachiara, SciPost Phys. 1, 009 (2016); arXiv:1607.07224], results for the four-point functions were obtained, based on the bootstrap approach, combined with simple conjectures for the spectra in the different fusion channels. In this paper, we test these conjectures using lattice algebraic considerations combined with extensive numerical studies of correlations on i…
We introduce a method to measure the entanglement entropy using a wavelet analysis. In the method we perform the two-dimensional Haar wavelet transform of configuration of Fortuin-Kasteleyn (FK) clusters. The configuration represents a direct snapshot of spin-spin correlations since spin degrees of freedom are traced out in FK representation. A snapshot of FK clusters loses image information at each coarse-graining process by the wavelet transform. We show that the loss of image information measures the entanglement entropy in the Potts model.
Due to Fortuin and Kastelyin the $q$ state Potts model has a representation as a sum over random graphs, generalizing the Potts model to arbitrary $q$ is based on this representation. A key element of the Random Cluster representation is the combinatorial factor $Γ_{\Graph{G}}(\Clusters,\Edges)$, which is the number of ways to form $\Clusters$ distinct clusters, consisting of totally $\Edges$ edges. We have devised a method to calculate $Γ_{\Graph{G}}(\Clusters,\Edges)$ from Monte Carlo simulations.
Fortuin-Kastelyn clusters in the critical $Q$-state Potts model are conformally invariant fractals. We obtain simulation results for the fractal dimension of the complete and external (accessible) hulls for Q=1, 2, 3, and 4, on clusters that wrap around a cylindrical system. We find excellent agreement between these results and theoretical predictions. We also obtain the probability distributions of the hull lengths and maximal heights of the clusters in this geometry and provide a conjecture for their form.
Recent results concerning the topological properties of random geometrical sets have been successfully applied to the study of the morphology of clusters in percolation theory. This approach provides an alternative way of inspecting the critical behaviour of random systems in statistical mechanics. For the 2d q-states Potts model with q <= 6, intensive and accurate numerics indicates that the average of the Euler characteristic (taken with respect to the Fortuin-Kasteleyn random cluster measure) is an order parameter of the phase transition.
These notes give examples of how suitably defined geometrical objects encode in their fractal structure thermal critical behavior. The emphasis is on the two-dimensional Potts model for which two types of spin clusters can be defined. Whereas the Fortuin-Kasteleyn clusters describe the standard critical behavior, the geometrical clusters describe the tricritical behavior that arises when including vacant sites in the pure Potts model. Other phase transitions that allow for a geometrical description discussed in these notes include the superfluid phase transition and Bose-Einstein condensation.
We present a technique, which we call "etching," which we use to study the harmonic measure of Fortuin-Kasteleyn clusters in the Q-state Potts model for Q=1-4. The harmonic measure is the probability distribution of random walkers diffusing onto the perimeter of a cluster. We use etching to study regions of clusters which are extremely unlikely to be hit by random walkers, having hitting probabilities down to 10^(-4600). We find good agreement between the theoretical predictions of Duplantier and our numerical results for the generalized dimension D(q), including regions of small and negative …
We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a $q$-state Potts model coupled to an $r$-state Potts model in two dimensions. For integer values of $q$ and $r$ the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin-Teller case ($q=r=2$). Since our results extend to continuous values of the number of states, we can access the limit $r\to 1$ corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot…
We have considered clusters of like spin in the Q-Potts model, the spin Potts clusters. Using Monte Carlo simulations, we studied these clusters on a square lattice with periodic boundary conditions for values of Q in [1,4]. We continue the work initiated with Delfino and Viti (2013) by measuring the universal finite size corrections of the two-point connectivity. The numerical data are perfectly compatible with the CFT prediction, thus supporting the existence of a consistent CFT, still unknown, describing the connectivity Potts spin clusters. We provided in particular new insights on the ene…
The invaded cluster approach is extended to 2D Potts model with annealed vacancies by using the random-cluster representation. Geometrical arguments are used to propose the algorithm which converges to the tricritical point in the two-dimensional parameter space spanned by temperature and the chemical potential of vacancies. The tricritical point is identified as a simultaneous onset of the percolation of a Fortuin-Kasteleyn cluster and of a percolation of "geometrical disorder cluster". The location of the tricritical point and the concentration of vacancies for q = 1, 2, 3 are found to be in…
We consider the fractal dimensions of critical clusters occurring in configurations of a q-state Potts model coupled to the planar random graphs of the dynamical triangulations formulation of Euclidean quantum gravity in two dimensions. For regular lattices, it is well-established that at criticality the properties of Fortuin-Kasteleyn clusters are directly related to the conventional critical exponents, whereas the corresponding properties of the geometric clusters of like spins are not. Recently it has been observed that the latter are related to the critical properties of a tricritical Pott…
We address the geometrical critical behavior of the two-dimensional Q-state Potts model in terms of the spin clusters (i.e., connected domains where the spin takes a constant value). These clusters are different from the usual Fortuin-Kasteleyn clusters, and are separated by domain walls that can cross and branch. We develop a transfer matrix technique enabling the formulation and numerical study of spin clusters even when Q is not an integer. We further identify geometrically the crossing events which give rise to conformal correlation functions. This leads to an infinite series of fundamenta…
A new algorithm is presented, which allows to calculate numerically the partition function Z_q of the d-dimensional q-state Potts models for arbitrary real values q>0 at any given temperature T with high precision. The basic idea is to measure the distribution of the number of connected components in the corresponding Fortuin-Kasteleyn representation and to compare with the distribution of the case q=1 (graph percolation), where the exact result Z_1=1 is known. As application, d=2 and d=3-dimensional ferromagnetic Potts models are studied, and the critical values q_c, where the transition chan…
We study the dynamic critical behavior of the local bond-update (Sweeny) dynamics for the Fortuin-Kasteleyn random-cluster model in dimensions d=2,3, by Monte Carlo simulation. We show that, for a suitable range of q values, the global observable S_2 exhibits "critical speeding-up": it decorrelates well on time scales much less than one sweep, so that the integrated autocorrelation time tends to zero as the critical point is approached. We also show that the dynamic critical exponent z_{exp} is very close (possibly equal) to the rigorous lower bound α/ν, and quite possibly smaller than the cor…
We study the chromatic polynomial P_G(q) for m \times n triangular-lattice strips of widths m <= 12_P, 9_F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin--Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n\to\infty. We recompute the limiting cur…
We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P_G(q) for m \times n rectangular subsets of the square lattice, with m \le 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n \to\infty, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers B_2,B_3…
We consider the random-bond Potts model in the large-$Q$ limit and calculate the excess entropy, $S_Γ$, of a contour, $Γ$, which is given by the mean number of Fortuin-Kasteleyn clusters which are crossed by $Γ$. In two dimensions $S_Γ$ is proportional to the length of $Γ$, to which - at the critical point - there are universal logarithmic corrections due to corners. These are calculated by applying techniques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess ent…
We investigate the operator content of the Q-state Potts model in arbitrary dimension, using the representation theory of the symmetric group. In particular we construct all possible tensors acting on N spins, corresponding to given symmetries under $S_Q$ and $S_N$, in terms of representations involving any Young diagram. These operators transform non-trivially under the group of spatial rotations, with a definite conformal spin. The two-point correlation functions are then computed, and their physical interpretation is given in terms of Fortuin-Kasteleyn clusters propagating between two neigh…
We study the chromatic polynomial P_G(q) for m \times n square- and triangular-lattice strips of widths 2\leq m \leq 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin--Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n\to\infty. We find that the different phases that appear in this model can be characterized by a topological…
We discuss a new cluster representation for the internal energy and the specific heat of the d-dimensional Ising model, obtained by studying the percolation mapping of an Ising model with an arbitrary set of antiferromagnetic links. Such a representation relates the thermal operators to the topological properties of the Fortuin-Kasteleyn clusters of Ising percolation and is a powerful tool to get new exact relations on the topological structure of FK clusters of the Ising model defined on an arbitrary graph.
We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z \ge α/νis close to but probably not sharp in d=2, and is far from sharp in d=3, for all q. The conjecture z \ge β/νis false (for some values of q) in both d=2 and d=3.
We revisit the problem of Q-colourings of the triangular lattice using a mapping onto an integrable spin-one model, which can be solved exactly using Bethe Ansatz techniques. In particular we focus on the low-energy excitations above the eigenlevel g_2, which was shown by Baxter to dominate the transfer matrix spectrum in the Fortuin-Kasteleyn (chromatic polynomial) representation for Q_0 <= Q <= 4, where Q_0 = 3.819671... We argue that g_2 and its scaling levels define a conformally invariant theory, the so-called regime IV, which provides the actual description of the (analytically continued…
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 $d_{\rm B} = 1.732 \, 0(3)$ 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…
We show how to use the link representation of the transfer matrix $D_N$ of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin-Kasteleyn model with various boundary conditions and parameter $β= 2 \cos(π(1-a/b)), a,b\in \mathbb N$ and, more specifically, partition functions of the corresponding $Q$-Potts spin models, with $Q=β^2$. The braid limit of $D_N$ is shown to be a central element $F_N(β)$ of the Temperley-Lieb algebra $TL_N(β)$, its eigenvalues are determined and, for generic $β$, a basis of its eigenvectors is constructed using the Wenzl-Jones pr…
Ising and Potts models can be studied using the Fortuin-Kasteleyn representation through the Edwards-Sokal coupling. This adapts to the setting where the models are exposed to an external field of strength $h>0$. In this representation, which is also known as the random-cluster model, the Kertész line separates the two regions of parameter space according to the existence of an infinite cluster in $\mathbb{Z}^d$. This signifies a geometric phase transition between the ordered and disordered phases even in cases where a thermodynamic phase transition does not occur. In this article, we prove st…
We study, via Monte Carlo simulation, the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts ferromagnet to non-integer q \ge 1. We consider spatial dimension d=2 and 1.25 \le q \le 4 in steps of 0.25, on lattices up to 1024^2, and obtain estimates for the dynamic critical exponent z_{CM}. We present evidence that when 1 \le q \lesssim 1.95 the Ossola-Sokal conjecture z_{CM} \ge β/νis violated, though we also present plausible fits compatible with this conjecture. We show tha…
Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering -- both in short-range (EA) and infinite-range (SK) models -- within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the $\pm J$ EA model in three dimensions and on rigorous results for the SK…
We consider two models with disorder dominated critical points and study the distribution of clusters which are confined in strips and touch one or both boundaries. For the classical random bond Potts model in the large-q limit we study optimal Fortuin-Kasteleyn clusters by combinatorial optimization algorithm. For the random transverse-field Ising chain clusters are defined and calculated through the strong disorder renormalization group method. The numerically calculated density profiles close to the boundaries are shown to follow scaling predictions. For the random bond Potts model we have …
Backbone three-point correlation function in the two-dimensional Potts model
Hefei University of Technology（合肥工业大学）；University of Science and Technology of China（中国科学技术大学）；CNRS-Laboratoire de Physique de l’Ecole Normale Supériure, PSL Research University, Sorbonne Université, Université Paris Cité（法国国家科学研究中心-巴黎高等师范学院物理实验室，PSL研究大学，索邦大学，巴黎西岱大学）；Institut de Physique Théorique, CEA, CRNS, Université Paris–Saclay（理论物理研究所，法国原子能和替代能源委员会，法国国家科学研究中心，巴黎萨克雷大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
The simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down. A number of cluster algorithms, usually based on the Fortuin-Kasteleyn representation of the Potts model, and suitable generalizations for continuous-spin models have been used to increase simulation efficiency. The first algorithm making use of this representation, suggested by Sweeny in 1983, has not found widespread adoption due to problems in its implementation. However, it has been recently shown that it is indeed more efficient in reducing …
We investigate the damage spreading effect in the Fortuin-Kasteleyn random cluster model for 2- and 3-dimensional grids with periodic boundary. For 2D the damage function has a global maximum at $p=\sqrt{q}/(1+\sqrt{q})$ for all $q>0$ and also local maxima at $p=1/2$ and $p=q/(1+q)$ for $q\lesssim 0.75$. For 3D we observe a local maximum at $p=q/(1+q)$ for $q\lesssim 0.46$ and a global maximum at $p=1/2$ for $q\lesssim 4.5$. The chaotic phase of the model's $(p,q)$-parameter space is where the coupling time is of exponential order and we locate points on its boundary. For 3-dimensional grids t…
