We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry $S_q$ in $d=6-ε$ (Landau-Potts field theories) and $d=4-ε$ (hypertetrahedral models) up to three loops.We use our results to determine the $ε$-expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests ($q\to0$), and bond percolations ($q\to1$). We also explicitly verify several expected degeneracies in the spectrum of relevant operators for natural values of $q$ upon analytic continuation, which are linked to logarithmic corrections of CFT correlators, and use the $ε$-expansion to determine the universal coefficients of such logarithms.
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.
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.
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 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$_κ$.
Monte Carlo (MC) and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the two-dimensional 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed and an approach is proposed in order to account numerically for these corrections in the determination of critical amplitudes. Accurate estimates of universal amplitude ratios $A_+/A_-$, $Γ_+/Γ_-$, $Γ_T/Γ_-$ and $R_C^\pm$ are given, which arouse new questions with respect to previous works.
We study the fractal properties of interfaces in the 2d Ashkin-Teller model. The fractal dimension of the symmetric interfaces is calculated along the critical line of the model in the interval between the Ising and the four-states Potts models. Using Schramm's formula for crossing probabilities we show that such interfaces can not be related to the simple SLE$_κ$, except for the Ising point. The same calculation on non-symmetric interfaces is performed at the four-states Potts model: the fractal dimension is compatible with the result coming from Schramm's formula, and we expect a simple SLE$…
Monte Carlo and series expansion data for the energy, specific heat, magnetisation and susceptibility of the 4-state Potts model in the vicinity of the critical point are analysed. The role of logarithmic corrections is discussed. Estimates of universal ratios A_+/A_-, $Γ_+/Γ_L$, $Γ_T/Γ_L$ and R_c^+ are given.
We study numerically the fractal dimensions and the bulk three-point connectivity for the spin clusters of the Q-state Potts model in two dimensions with $1\leq Q\leq 4$. We check that the usually invoked correspondence between FK clusters and spin clusters works at the level of fractal dimensions. However, the fine structure of the conformal field theories describing critical clusters first manifests at the level of the three-point connectivities. Contrary to what recently found for FK clusters, no obvious relation emerges for generic Q between the spin cluster connectivity and the structure …
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.
Critical systems have always intrigued physicists and precipitated the development of new techniques. Recently, there has been renewed interest in the information contained in their classical configurations, whose computation do not require full knowledge of the wavefunction. Inspired by holographic duality, we investigated the entanglement properties of the classical configurations (snapshots) of the Potts model by introducing an ansatz ensemble of random fractal images. By virtue of the central limit theorem, our ansatz accurately reproduces the entanglement spectra of actual Potts snapshots…
Using the symmetric group $S_Q$ symmetry of the $Q$-state Potts model, we classify the (scalar) operator content of its underlying field theory in arbitrary dimension. In addition to the usual identity, energy and magnetization operators, we find fields that generalize the $N$-cluster operators well-known in two dimensions, together with their subleading counterparts. We give the explicit form of all these operators -- up to non-universal constants -- both on the lattice and in the continuum limit for the Landau theory. We compute exactly their two- and three-point correlation functions on an …
The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a $Q$ -state Potts cluster, is solved in two dimensions. The dimension $\hat f(θ)$ of the boundary set with local wedge angle $θ$ is $\hat f(θ)=\fracπθ -\frac{25-c}{12} \frac{(π-θ)^2}{θ(2π-θ)}$, with $c$ the central charge of the model. As a corollary, the dimensions $D_{\rm EP} =sup_θ\hat f(θ)$ of the external perimeter and $D_{\rm H}$ of the hull of a Potts cluster obey the duality equation $(D_{\rm EP}-1)(D_{\rm H}-1)={1/4}$. A related covariant MF …
We study the critical behavior of the q-state Potts model with random ferromagnetic couplings. Working with the cluster representation the partition sum of the model in the large-q limit is dominated by a single graph, the fractal properties of which are related to the critical singularities of the random Potts model. The optimization problem of finding the dominant graph, is studied on the square lattice by simulated annealing and by a combinatorial algorithm. Critical exponents of the magnetization and the correlation length are estimated and conformal predictions are compared with numerical…
We present a solution of the non-linear renormalization group equations leading to the dominant and subdominant singular behaviours of physical quantities (free energy density, correlation length, internal energy, specific heat, magnetization, susceptibility and magnetocaloric coefficient) at the critical temperature in a non- vanishing magnetic field. The solutions i) lead to exact cancellation of logarithmic corrections in universal amplitude ratios and ii) prove recently proposed relations among logarithmic exponents.
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…
The q-state ferromagnetic Potts model under a non-zero magnetic field coupled with the 0^th Potts state was investigated by an exact real-space renormalization group approach. The model was defined on a family of diamond hierarchical lattices of several fractal dimensions d_F. On these lattices, the renormalization group transformations became exact for such a model when a correlation coupling that singles out the 0^th Potts state was included in the Hamiltonian. The rich criticality presented by the model with q=3 and d_F=2 was fully analyzed. Apart from the Potts criticality for the zero fie…
We develop a field-theoretic representation for the configurations of an interface between two ordered phases of a q-state Potts model in two dimensions, in the solid-on-solid approximation. The model resembles the field theory of directed percolation and may be analysed using similar renormalisation group methods. In the one-loop approximation these reveal a simple mechanism for the emergence of a critical value q_c, such that for q<q_c the interface becomes a fractal with a vanishing interfacial tension at the critical point, while for q>q_c the interfacial width diverges at a finite value o…
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…
Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q = 1 limit of the Q-state Potts model, and analyzing the underlying S_Q symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q = 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates tw…
We develop a dynamic field-theoretic renormalization-group (RG) theory for the cooling first-order phase transitions in the Potts model. It is suggested that the well-known imaginary fixed points of the $q$-state Potts model for $q>10/3$ in the RG theory are the origin of the dynamic scaling found recently, apart from the logarithmic corrections. This indicates that the real and imaginary fixed points of the Potts model are both physical and control the scalings of the continuous and discontinuous phase transitions, respectively, of the model. Our one-loop results for the scaling exponents are…
We study the kinetics of the two-dimensional q > 4-state Potts model after a shallow quench slightly below the critical temperature and above the pseudo spinodal. We use numerical methods and we focus on intermediate values of q, 4 < q < 100. We show that, initially, the system evolves as if it were quenched to the critical temperature. The further decay from the metastable state occurs by nucleation of k out of the q possible phases. For a given quench temperature, k is a logarithmically increasing function of the system size. This unusual finite size dependence is a consequence of a scaling …
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…
We present measurements of the fractal dimensions associated to the geometrical clusters for Z_4 and Z_5 spin models. We also attempted to measure similar fractal dimensions for the generalised Fortuyin Kastelyn (FK) clusters in these models but we discovered that these clusters do not percolate at the critical point of the model under consideration. These results clearly mark a difference in the behaviour of these non local objects compared to the Ising model or the 3-state Potts model which corresponds to the simplest cases of Z_N spin models with N=2 and N=3 respectively. We compare these f…
We use the single-histogram technique to study the critical behavior of the three-state Potts model on a (random) Voronoi-Delaunay lattice with size ranging from 250 to 8000 sites. We consider the effect of an exponential decay of the interactions with the distance,$J(r)=J_0\exp(-ar)$, with $a>0$, and observe that this system seems to have critical exponents $γ$ and $ν$ which are different from the respective exponents of the three-state Potts model on a regular square lattice. However, the ratio $γ/ν$ remains essentially the same. We find numerical evidences (although not conclusive, due to t…
Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios …
We study the critical behavior of the random q-state Potts model in the large-q limit on the diamond hierarchical lattice with an effective dimensionality $d_{\rm eff} > 2$. By varying the temperature and the strength of the frustration the system has a phase transition line between the paramagnetic and the ferromagnetic phases which is controlled by four different fixed points. According to our renormalization group study the phase-boundary in the vicinity of the multicritical point is self-similar, it is well represent ed by a logarithmic spiral. We expect infinite number of reentrances in t…
We derive, from conformal invariance and quantum gravity, the multifractal spectrum f(alpha,c) of the harmonic measure (or electrostatic potential, or diffusion field) near any conformally invariant fractal in two dimensions, corresponding to a conformal field theory of central charge c. It gives the Hausdorff dimension of the set of boundary points where the potential varies with distance r to the fractal frontier as r^{alpha}. First examples are a Brownian frontier, a self-avoiding walk, or a percolation cluster. Potts, O(N) models, and the so-called SLE process are also considered. Higher m…
Boltzmann's microcanonical entropy is the link between statistical physics and thermodynamics, forasmuch as the behavior of any thermodynamic quantity is directly related to the number of microscopic configurations. Accordingly, in this work, we investigate the behavior of the logarithm of the density of states of the three-state Potts model with an external field applied to one of the states using joint entropic simulations based on the Wang-Landau algorithm. Our analysis reveals that the microcanonical entropy curve is detachable, and each resulting path is related to the formation of cluste…
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）
We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n->0 of the O(n) model, …
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 $F$ coupled $q$-state Potts models in a two-dimensional square lattice. The interaction between the different layers is attractive, to favour a simultaneous alignment in all of them, and its strength is fixed. The nature of the phase transition for zero field is numerically determined for $F=2,3$. Using the Lee-Kosterlitz method, we find that it is continuous for $F=2$ and $q=2$, whereas it is abrupt for higher values of $q$ and/or $F$. When a continuous or a weakly first-order phase transition takes place, we also analyze the properties of the geometrical clusters. This allows us to …
Evolving fractal dimensions in iterative bicolored percolation
University of Science and Technology of China（中国科学技术大学）；Peking University（北京大学）；Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China（中国科学技术大学微尺度物质科学国家研究中心）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）；Hefei University of Technology（合肥工业大学）
We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get $y_{h1}=0.780(3)$, $η_\parallel=1.44(1)$, and $η_\perp=0.736(6)$; for the special transition, we get $y_s=0.59(1)$, $y_{h1}=1.693(2)$, $η_\parallel=-0.391(4)$, and $η_\perp=-0.179(5)$; in the extraordinary-log phase, the surface correlation funct…
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…
The Potts model is one of the most popular spin models of statistical physics. The prevailing majority of work done so far corresponds to the lattice version of the model. However, many natural or man-made systems are much better described by the topology of a network. We consider the q-state Potts model on an uncorrelated scale-free network for which the node-degree distribution manifests a power-law decay governed by the exponent λ. We work within the mean-field approximation, since for systems on random uncorrelated scale-free networks this method is known to often give asymptotically exact…
The nonequilibrium critical dynamics of the Ising magnet on a fractal substrate, namely the Sierpinski carpet with Hausdorff dimension $d_H$ =1.7925, has been studied within the short-time regime by means of Monte Carlo simulations. The evolution of the physical observables was followed at criticality, after both annealing ordered spin configurations (ground state) and quenching disordered initial configurations (high temperature state), for three segmentation steps of the fractal. The topological effects become evident from the emergence of a logarithmic periodic oscillation superimposed to a…
We study phase-ordering dynamics of a ferromagnetic system with a scalar order-parameter on fractal graphs. We propose a scaling approach, inspired by renormalization group ideas, where a crossover between distinct dynamical behaviors is induced by the presence of a length $λ$ associated to the topological properties of the graph. The transition between the early and the asymptotic stage is observed when the typical size $L(t)$ of the growing ordered domains reaches the crossover length $λ$. We consider two classes of inhomogeneous substrates, with different activated processes, where the effe…
We study the dynamic critical behavior of two algorithms: the Swendsen-Wang algorithm for the two-dimensional Potts model with q=2,3,4 and a Swendsen-Wang-type algorithm for the two-dimensional symmetric Ashkin-Teller model on the self-dual curve. We find that the Li--Sokal bound on the autocorrelation time τ_{{\rm int},{\cal E}} \geq const \times C_H is almost, but not quite sharp. The ratio τ_{{\rm int},{\cal E}}/C_H appears to tend to infinity either as a logarithm or as a small power (0.05 \ltapprox p \ltapprox 0.12). We also show that the exponential autocorrelation time τ_{{\rm exp},{\ca…
We study the quantum Ising model on the Sierpiński triangle, whose Hausdorff dimension is $\log 3/ \log 2 \approx 1.585$, and demonstrate that it undergoes second-order phase transition with scaling relations satisfied precisely. We also study the quantum $3$-state Potts model on the Sierpiński triangle and find first-order phase transition, which is consistent with a prediction from $ε$-expansion that the transition becomes first-order for $D > 1.3$. We then compute critical exponents of the Ising model on higher-dimensional Sierpiński pyramids with various Hausdorff dimension via Monte-Carlo…
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 dif…
The phase transition in the q-state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a pice-wise linear function of the temperature, which is rounded after averaging…
Monte Carlo (MC) simulations and series expansions (SE) data for the energy, specific heat, magnetization, and susceptibility of the three-state and four-state Potts model and Baxter-Wu model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\le …
Percolation, a paradigmatic geometric system in various branches of physical sciences, is known to possess logarithmic factors in its correlators. Starting from its definition, as the $Q\rightarrow1$ limit of the $Q$-state Potts model with $S_Q$ symmetry, in terms of geometrical clusters, its operator content as $N$-cluster observables has been classified. We extensively simulate critical bond percolation in two and three dimensions and determine with high precision the $N$-cluster exponents and non-scalar features up to $N \! =\! 4$ (2D) and $N \! =\! 3$ (3D). The results are in excellent agr…
Tensor Renormalization Group Calculations of Partition-Function Ratios
Graduate School of Science and Technology, Keio University（庆应义塾大学理工学部）；Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University（庆应义塾大学可持续量子人工智能中心）；Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）；Trans-scale Quantum Science Institute, The University of Tokyo（东京大学跨尺度量子科学研究所）
Using Monte Carlo simulations we study the distributions of the 3-block mass $N_3$ in 4d, 5d, and 6d percolation systems. Because the probability of creating large 3-blocks in these dimensions is very small, we use a ``go with the winners'' method of statistical enhancement to simulate configurations having probability as small as $10^{-30}$. In earlier work, the fractal dimensions of 3-blocks, $d_3$, in 2d and 3d were found to be $1.20\pm 0.1$ and $1.15\pm 0.1$, respectively, consistent with the possibility that the fractal dimension might be the same in all dimensions. We find that the fract…
The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is beta = -2cos(pi/barkappa}) where the parameter barkappa is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges c = 13 - 6(barkappa + barkappa^{-1}) and, for the minimal models of interest here, barkapp…
In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to value…
The spectrum of conformal weights for the CFT describing the two-dimensional critical $Q$-state Potts model (or its close cousin, the dense loop model) has been known for more than 30 years. However, the exact nature of the corresponding $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ representations has remained unknown up to now. Here, we solve the problem for generic values of $Q$. This is achieved by a mixture of different techniques: a careful study of "Koo--Saleur generators" [arXiv:hep-th/9312156], combined with measurements of four-point amplitudes, on the numerical side, and OPEs and the fou…
We construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two, in the case of zero external magnetic field, using a generalization of the combinatorial method of Feynman and Vodvickenko. Our procedure is applicable to any fractal obtained by the removal of sites of a periodic two dimensional lattice. As a first application, we compute estimates for the critical temperatures of many different Sierpinski carpets and we compare them to known Monte Carlo estimates. The results show that our method is capable of determining the critical …
We present a systematic study of the nested sampling algorithm based on the example of the Potts model. This model, which exhibits a first order phase transition for $q>4$, exemplifies a generic numerical challenge in statistical physics: The evaluation of the partition function and thermodynamic observables, which involve high dimensional sums of sharply structured multi-modal density functions. It poses a major challenge to most standard numerical techniques, such as Markov Chain Monte Carlo. In this paper we will demonstrate that nested sampling is particularly suited for such problems and …
This article gives a comprehensive description of the fractal geometry of conformally-invariant (CI) scaling curves, in the plane or half-plane. It focuses on deriving critical exponents associated with interacting random paths, by exploiting an underlying quantum gravity (QG) structure, which uses KPZ maps relating exponents in the plane to those on a random lattice, i.e., in a fluctuating metric. This is applied to critical models, like O(N) and Potts models, and to the Stochastic Löwner Evolution (SLE). The multifractal (MF) function f(alpha, c) of the harmonic measure near any CI fractal b…
The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of th…
Fractals of Simple Random Walks in Two Dimensions: A Monte Carlo Study
Department of Physics, Guizhou University（贵州大学物理系）；Department of Information Science and Technology, Donghua University（东华大学信息科学与技术学院）；Hefei National Laboratory（合肥国家实验室）
We study the transport properties of directed percolation clusters at the upper critical dimension $d_{c} = 4+1$, where critical fluctuations induce logarithmic corrections to the leading (mean-field) scaling behavior. Employing field theory and renormalization group methods we calculate these logarithmic corrections up to and including the next to leading correction for a variety of observables, viz. the connectivity, i.e., the probability that two given points are connected, the average two-point resistance and some of the fractal masses describing percolation clusters. Furthermore, we study…
