Percolation of dimers on square lattices

A theoretical approach, based on exact calculations of configurations on finite rectangular cells, is applied to study the percolation of homonuclear dimers on square lattices. An efficient algorithm allows us to calculate the detailed structure of the configuration space for M=_Lx×_Ly$M=L_x\times L_y$ cells, with M$M$ varying from 16 to 36. The percolation process has been monitored by following the percolation function, defined as the ratio between the number of percolating configurations and the total number of available configurations for a given cell size and concentration of occupied sites. The percolation threshold has been calculated by means of two complementary methods: one based on well-known renormalization techniques and the other based on determining the inflection point of the percolation function curves. A comparison of the results obtained by these two methods has been performed. The study includes the use of finite-size scaling theory to extrapolate numerical results towards the thermodynamic limit. The effect of jamming due to dimers is also established. Finally, the critical exponents ν$\nu$, β$\beta$ and γ$\gamma$ have been obtained and values compared with numerical results and expected theoretical estimations. The present results show agreement and even improvement (in the case of γ$\gamma$) with respect to some numeric values available in the literature.     

Scaling inequalities for oriented percolation

We look at seven critical exponents associated with two-dimensional oriented percolation. Scaling theory implies that these quantities satisfy four equalities. We prove five related inequalitites.     

Height-height correlations for surface growth on percolation networks

The height-height correlations of the surface growth for equilibrium and nonequilibrium restricted solid-on-solid (RSOS) model were investigated on randomly diluted lattices, i.e., on infinite percolation networks. It was found that the correlation function calculated over the chemical distances reflected the dynamics better than that calculated over the geometrical distances. For the equilibrium growth on a critical percolation network, the correlation function for the evolution time t?1 yielded a power-law behavior with the power ζ^′, associated with the roughness exponent ζ via the relation ζ=ζ^′d_f/d_l, with d_f and d_l being, respectively, the fractal dimension and the chemical dimension of the substrate. For the nonequilibrium growth, on the other hand, the correlation functions did not yield power-law behaviors for the concentration of diluted sites x less than or equal to the critical concentration x_c.     

Measures of critical exponents in the four-dimensional site percolation

Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-to-scaling exponent and, with great accuracy, the critical density.     

Multifractality in elastic percolation

We present numerical results on the distribution of forces in the central-force percolation model at threshold in two dimensions. We conjecture a relation between the multifractal spectrum of scalar and vector percolation that we test for central-foce percolation. This relation is in excellent agreement with our numerical data.     

Gap exponents for percolation processes with triangle condition

A study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region. It is show that the gaps are given by _t =2 fort=2, 3,. Scaling theory predicts thatP _p (¦C ₀¦S(p))–(p _c –p) andE _p (1/¦C ₀¦; ¦C ₀¦S(p))–(p _c –p)³, whereS(p) is the typical cluster size. It is found that (p _c –p)P _p (|C ₀S(p) ^1–)(p _c –p)^1–2 and (p _c –p)³E _p (1/|C ₀|;|C ₀|S(p) ^1–))(p _c –p)^3–4.     

Corrections to scaling for diffusion exponents on three-dimensional percolation systems at criticality

Recent results of Monte Carlo simulations of the ant-in-the-labyrinth method in three-dimensional percolation lattices are reanalyzed in the light of more accurate corrections to scaling ansatz, motivated by inconsistent results that have appeared in the literature. The results are observed to be sensitive to the form of the scaling correction terms. Using a single correction term, we estimate the valuek=0.197±0.004 for the anomalous diffusion exponent at criticality. When two correction terms are included,k=0.200±0.002 is obtained. These new estimates are consistent with known theoretical bounds, with recent series expansion results, and with numerical calculations of the conductance of random resistor networks above criticality.     

Molecular trajectory algorithm for random walks on percolation systems at criticality in two and three dimensions

Random walk simulations based on a molecular trajectory algorithm are performed on critical percolation clusters. The analysis of corrections to scaling is carried out. It has been found that the fractal dimension of the random walk on the incipient infinite cluster is d_w=2.873±0.008 in two dimensions and 3.78 ± 0.02 in three dimensions. If instead the diffusion is averaged over all clusters at the threshold not subject to the infinite restriction, the corresponding critical exponent k is found to be k=0.3307±0.0014 for two-dimensional space and 0.199 ± 0.002 for three-dimensional space. Moreover, in our simulations the asymptotic behaviors of local critical exponents are reached much earlier than in other numerical methods.     

Phase structure of two-dimensional spin models and percolation

For a class of classical spin models in 2D satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially. The class contains discrete and continuous spin systems with Abelian and non-Abelian symmetry groups. For the discrete models our results imply that they show either long-range order or are in a soft phase characterized by powerlike decay of correlations; for the continuous models only the second possibility exists. The continuous models include a version of the plane rotator [O(2)] model; for this model we rederive, modulo two conjectures, the Fröhlich-Spencer result on the existence of the Kosterlitz-Thouless phase in a very simple way. The proof is based on percolation-theoretic and topological arguments.     

Typical cluster size for two-dimensional percolation processes

The typical cluster size for two-dimensional percolation models is discussed. It is shown that, forW ₀={xZ ²0x}, [–lim _n(1/n) logP _p (W ₀=n)]^–1p–p _c ^– aspp _c , provided thatE _p (W ₀²)/E _p (W ₀)p–P _c ^– aspp _c . Furthermore, we introduce a new quantityf _s (p), which may be thought of as the singular part of the free energy, and show thatf _s (p)¦p–p _c ¦^2v provided that the correlation length ¦p–p _c ¦^–v aspp _c .     

Monte Carlo study of percolation on disordered triangular lattices

A simple model for amorphous solids, consisting of a mixed bond triangular lattice with a fraction of attenuated bonds randomly distributed (which simulate the presence of defects in the surface), is studied here by using computational simulation. The degree of disorder of the surface is tunable by selecting the values of (1) the fraction of regular [attenuated] bonds ρ [1−ρ] (0≤ρ≤1) and (2) the factor r, which is defined as the ratio between the value of the conductivity associated to an attenuated bond and that corresponding to a regular bond (0≤r≤1). The results obtained show how the percolation properties of the disordered system are modified with respect to the standard random bond percolation problem (r=0).     

Bond percolation with parallelism restriction on square lattices

As a simple approximation for the ±J spin glass we studied bond percolation on square lattices. However, two neighboring chains of ferromagnetic bonds are required for spins to be regarded as connected. We determine the percolation thresholdp _c =0.8282±0.0002 and the critical exponent =0.75 _–0.05 ^+0.02 for this specific percolation by means of Monte-Carlo simulation on square lattices (up to 150×150).     

Another critical exponent inequality for percolation: β⩾2/δ

The inequality in the title is derived for standard site percolation in any dimension, assuming only that the percolation density vanishes at the critical point. The proof, based on a lattice animal expansion, is fairly simple and is applicable to rather general (site or bond, short-or long-range) independent percolation models.     

Continuum percolation conductivity exponents in restricted domains

Conductivity behavior of continuum percolation in restricted two-dimensional domains is simulated by considering systems of randomly distributed disks. The domain is restricted in that conducting objects are permitted to lie in only a portion of the domain. Such a restricted domain might better approximate some natural systems. Simulations of two-dimensional systems, based on three distributions of local conductances, are examined and found to demonstrate a power-law behavior with conductivity exponents smaller than those arising in regular lattice and continuum percolation     

Strict inequalities for some critical exponents in two-dimensional percolation

For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL():={(x, y):x=r cos, y=r sin for some r0, or} and():=limpp _c [log(p–p _c )]^–1 log P_cr {itO is connected to by an occupied path inL()}, then() is strictly decreasing in on [0, 2]. Similarly, lim_n [–logn]^–1 logP _cr {itO is connected by an occupied path inL()() to the exterior of [–n, n]×[–n, n] is strictly decreasing in on [0, 2].     

Percolation Phenomena in Low and High Density Systems

We consider the 2D quenched–disordered q–state Potts ferromagnets and show that in the translation invariant measure, averaged over the disorder, at self–dual points any amalgamation of q?1 species will fail to percolate despite an overall (high) density of 1?q ^?1. Further, in the dilute bond version of these systems, if the system is just above threshold, then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density. Finally, we demonstrate both phenomena in a single model by considering a “perturbation” of the dilute model that has a self–dual point. We also demonstrate that these phenomena occur, by a similar mechanism, in a simple coloring model invented by O. Häggström.     

Diffusion of lattice gases without double occupancy on three-dimensional percolation lattices

We examined the diffusion of lattice gases, where double occupancy of sites is excluded, on three-dimensional percolation lattices at the percolation thresholdp _c . The critical exponent for the root-mean-square displacement was determined to bek=0.183±0.010, which is similiar to the result of Roman for the problem of the ant in the labyrinth. Furthermore, we found a plateau value fork at intermediate times for systems with higher concentrations of lattice gas particles.     

Scaling analysis of the magnetocaloric effect in Gd5Si2Ge1.9X0.1 (X=Al, Cu, Ga, Mn, Fe, Co)

The field dependence of the magnetic entropy change has been studied for a series of doped Gd₅Si₂Ge₂ alloys, which possess a magnetic phase transition that is either entirely second order or a combination of primarily second-order mixed to a very minor degree with a first-order transition arising from a magneto-structural phase change. By analyzing the field scaling of the refrigerant capacity as well as of the reference temperatures used for constructing a universal scaling curve, a procedure for estimating the values of the critical exponents for the alloys was developed. For the cases where the transition is entirely second order, the results obtained from this procedure are comparable to the values obtained from the Kouvel–Fisher method. For the case of Fe-doped alloys which partially possess a first-order phase change, the Kouvel–Fisher method is inapplicable. However, their critical exponents determined by our developed procedure can be used to estimate the Curie temperature of the orthorhombic majority phase.     

Potts ferromagnet: Transformations and critical exponents in planar hierarchical lattices

We prove that the duality transformation for a Potts ferromagnet on two-rooted planar hierarchical lattices (HL) preserves the thermal eigenvalue. This leads to a relation between the correlation length critical exponents of a HL and its corresponding dual lattice. Using hyperscaling, we show that their specific heat critical exponents coincide. For a smaller class of HL—namely of diamond and tress types—we prove that another transformation also preserves and .     

A fluorescence study on the critical exponents for the linear polymerization of butyl-methacrylate

The steady-state fluorescence (SSF) technique was used to study the sol-gel transition for the linear bulk polymerization of butyl methacrylate (BMA), carried out above the glass transition temperature of polybutylmethacrylate (PBMA) (T _g?=?20°C). Pyrene (P_y) was used as the fluorescence probe. The increase in P_y intensity was monitored during free radical polymerization of BMA by using SSF technique. Changes in the viscosity of the pregel solutions due to gel formation dramatically enhance the fluorescent yield of aromatic molecules. This effect is used to monitor the sol-gel transition of BMA, as a function of time, at various temperatures. The results are interpreted in the view of percolation theory. The gel fraction exponent β?=?0.39?±?0.02 agreed the best with the static percolation values for the linear bulk BMA polymerization carried out above T _g but weight average degree of polymerization exponent,?γ?deviated from the percolation results.