Histogram Monte Carlo renormalization-group method for percolation problems

High-precision anomalous dimension of three-dimensional percolation and spatial profile of the critical giant cluster

In three-dimensional percolation, we apply and test the critical geometry approach for bounded critical phenomena based on the fractional Yamabe equation. The method predicts the functional shape of the order parameter profile ϕ, which is obtained by raising the solution of the Yamabe equation to the scaling dimension Δ_{ϕ}. The latter can be fixed from outcomes of numerical simulations, from which we obtain Δ_{ϕ}=0.47846(71) and the corresponding value of the anomalous dimension η=-0.0431(14). The comparison with values of η determined by using scaling relations is discussed. A test of hyperscaling is also performed.

Exact solution of the dimer model on the generalized finite checkerboard lattice

We present the exact closed-form expression for the partition function of a dimer model on a generalized finite checkerboard rectangular lattice under periodic boundary conditions. We investigate three different sets of dimer weights, each with different critical behaviors. We then consider different limits for the model on the three lattices. In one limit, the model for each of the three lattices is reduced to the dimer model on a rectangular lattice, which belongs to the c=-2 universality class. In another limit, two of the lattices reduce to the anisotropic Kasteleyn model on a honeycomb lattice, the universality class of which is given by c=1. The result that the dimer model on a generalized checkerboard rectangular lattice can manifest different critical behaviors is consistent with early studies in the thermodynamic limit and also provides insight into corrections to scaling arising from the finite-size versions of the model.

Scaling of cluster heterogeneity in percolation transitions

We investigate a critical scaling law for the cluster heterogeneity H in site and bond percolations in d-dimensional lattices with d = 2,...,6. The cluster heterogeneity is defined as the number of distinct cluster sizes. As an occupation probability p increases, the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase, and back to a monodisperse one in the supercritical phase. We show analytically that H diverges algebraically, approaching the percolation critical point p(c) as H |p-p(c)|(-1/σ) with the critical exponent σ associated with the characteristic cluster size. Interestingly, its finite-size-scaling behavior is governed by a new exponent ν H = 1+d (f)/(d)ν, where d(f) is the fractal dimension of the critical percolating cluster and ν is the correlation length exponent. The corresponding scaling variable defines a singular path to the critical point. All results are confirmed by numerical simulations.

Mapping functions and critical behavior of percolation on rectangular domains

The existence probability E_{p} and the percolation probability P of bond percolation on rectangular domains with different aspect ratios R are studied via the mapping functions between systems with different aspect ratios. The superscaling behavior of E_{p} and P for such systems with exponents a and b , respectively, found by Watanabe [Phys. Rev. Lett. 93, 190601 (2004)] can be understood from the lower-order approximation of the mapping functions f_{R} and g_{R} for E_{p} and P , respectively; the exponents a and b can be obtained from numerically determined mapping functions f_{R} and g_{R} , respectively.

Finite-size effects for the Ising model on helical tori

We analyze the exact partition function of the Ising model on a square lattice under helical boundary conditions obtained by Liaw [Phys. Rev. E 73, 055101(R) (2006)]. Based on such an expression, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point. From such results, we find that the shift exponent for the specific heat is lambda=1 for all values of the helicity factor d . We also find that finite-size corrections for the free energy, the internal energy, and the specific heat of the model in a crucial way depend on the helicity factor of the lattice.

Quasispecies theory for multiple-peak fitness landscapes

We use a path integral representation to solve the Eigen and Crow-Kimura molecular evolution models for the case of multiple fitness peaks with arbitrary fitness and degradation functions. In the general case, we find that the solution to these molecular evolution models can be written as the optimum of a fitness function, with constraints enforced by Lagrange multipliers and with a term accounting for the entropy of the spreading population in sequence space. The results for the Eigen model are applied to consider virus or cancer proliferation under the control of drugs or the immune system.

Efficient combination of Wang–Landau and transition matrix Monte Carlo methods for protein simulations

An efficient combination of the Wang-Landau and transition matrix Monte Carlo methods for protein and peptide simulations is described. At the initial stage of simulation the algorithm behaves like the Wang-Landau algorithm, allowing to sample the entire interval of energies, and at the later stages, it behaves like transition matrix Monte Carlo method and has significantly lower statistical errors. This combination allows to achieve fast convergence to the correct values of density of states. We propose that the violation of TTT identities may serve as a qualitative criterion to check the convergence of density of states. The simulation process can be parallelized by cutting the entire interval of simulation into subintervals. The violation of ergodicity in this case is discussed. We test the algorithm on a set of peptides of different lengths and observe good statistical convergent properties for the density of states. We believe that the method is of general nature and can be used for simulations of other systems with either discrete or continuous energy spectrum.

Superscaling of percolation on rectangular domains

For percolation on (RL)xL two-dimensional rectangular domains with a width L and aspect ratio R, we propose that the existence probability of the percolating cluster E(p)(L,epsilon,R) as a function of L, R, and deviation from the critical point epsilon can be expressed as F(epsilonL(y(t))R(a)), where y(t) identical with1/nu is the thermal scaling power, a is a new exponent, and F is a scaling function. We use Monte Carlo simulation of bond percolation on square lattices to test our proposal and find that it is well satisfied with a=0.14(1) for R>2. We also propose superscaling for other critical quantities.

Universal finite-size scaling functions with exact nonuniversal metric factors

Using exact partition functions and finite-size corrections for the Ising model on finite square, plane triangular, and honeycomb lattices and extending a method [J. Phys. 19, L1215 (1986)] to subtract leading singular terms from the free energy, we obtain universal finite-size scaling functions for the specific heat, internal energy, and free energy of the Ising model on these lattices with exact nonuniversal metric factors.

Exact finite-size corrections of the free energy for the square lattice dimer model under different boundary conditions

We express the partition functions of the dimer model on finite square lattices under five different boundary conditions (free, cylindrical, toroidal, Möbius strip, and Klein bottle) obtained by others (Kasteleyn, Temperley and Fisher, McCoy and Wu, Brankov and Priezzhev, and Lu and Wu) in terms of the partition functions with twisted boundary conditions Z(alpha, beta) with (alpha, beta)=(1/2,0), (0,1/2) and (1/2,1/2). Based on such expressions, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above. We find that the aspect-ratio dependence of finite-size corrections is sensitive to boundary conditions and the parity of the number of lattice sites along the lattice axis. We have also established several groups of identities relating dimer partition functions for the different boundary conditions.

Renormalization-group approach to an Abelian sandpile model on planar lattices

One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.

Convergence of threshold estimates for two-dimensional percolation

Using a recently introduced algorithm for simulating percolation in microcanonical (fixed-occupancy) samples, we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary, specifically for site percolation on a square lattice. We show that the convergence of the average-probability estimate is described by a nontrivial correction-to-scaling exponent as predicted previously, and measure the value of this exponent to be 0.90+/-0.02. For the median and cell-to-cell estimates of the percolation threshold we verify that convergence does not depend on this exponent, having instead a slightly faster convergence with a trivial analytic leading exponent.

Random-cluster multihistogram sampling for the q-state Potts model

Using the random-cluster representation of the q-state Potts models we consider the pooling of data from cluster-update Monte Carlo simulations for different thermal couplings K and number of states per spin q. Proper combination of histograms allows for the evaluation of thermal averages in a broad range of K and q values, including noninteger values of q. Due to restrictions in the sampling process correct normalization of the combined histogram data is nontrivial. We discuss the different possibilities and analyze their respective ranges of applicability.

Fast Monte Carlo algorithm for site or bond percolation

We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in percolation theory, including the position of the percolation transition for site percolation on the square lattice, the stretched exponential behavior of spanning probabilities away from the critical point, and the size of the giant component for site percolation on random graphs.

Exact universal amplitude ratios for two-dimensional Ising models and a quantum spin chain

Let f(N) and xi(-1)(N) represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a N x infinity lattice, with f(N)-->f(infinity) as N-->infinity. We obtain analytic expressions for a(k) and b(k) in the expansions N( f(N)-f(infinity)) = SUM (k = 1)(infinity)a(k)/N(2k-1) and xi(-1)(N) = SUM (k = 1)(infinity)b(k)/N(2k-1) for square, honeycomb, and plane-triangular lattices, and find that b(k)/a(k) = (2(2k)-1)/(2(2k-1)-1) for all of these lattices, i.e., the amplitude ratio b(k)/a(k) is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.

Efficient Monte Carlo algorithm and high-precision results for percolation

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at p(c) = 0.592 746 21(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.

Geometry, thermodynamics, and finite-size corrections in the critical Potts model

We establish an intriguing connection between geometry and thermodynamics in the critical q-state Potts model on two-dimensional lattices, using the q-state bond-correlated percolation model (QBCPM) representation. We find that the number of clusters <N(c)> of the QBCPM has an energy-like singularity for q not equal to 1, which is reached and supported by exact results, numerical simulation, and scaling arguments. We also establish that the finite-size correction to the number of bonds, <N(b)>, has no constant term and explains the divergence of related quantities as q-->4, the multicritical point. Similar analyses are applicable to a variety of other systems.

Cluster analysis and finite-size scaling for Ising spin systems

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction (c) of lattice sites in percolating clusters in subgraphs with n percolating clusters, f(n)(c), and the distribution function for magnetization (m) in subgraphs with n percolating clusters, p(n)(m). We find that f(n)(c) and p(n)(m) have very good finite-size scaling behavior and that they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions 1:square root[3]/2:square root[3]. The complex structure of the magnetization distribution function p(m) for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such a system.