Error estimation and reduction with cross correlations

Geometric clusters in the overlap of the Ising model

We study the percolation properties of geometrical clusters defined in the overlap space of two statistically independent replicas of a square-lattice Ising model that are simulated at the same temperature. In particular, we consider two distinct types of clusters in the overlap, which we dub soft- and hard-constraint clusters, and which are subsets of the regions of constant spin overlap. By means of Monte Carlo simulations and a finite-size scaling analysis we estimate the transition temperature as well as the set of critical exponents characterizing the percolation transitions undergone by these two cluster types. The results suggest that both soft- and hard-constraint clusters percolate at the critical temperature of the Ising model and their critical behavior is governed by the correlation-length exponent ν=1 found by Onsager. At the same time, they exhibit nonstandard and distinct sets of exponents for the average cluster size and percolation strength.

Two-dimensional dilute Baxter-Wu model: Transition order and universality

We investigate the critical behavior of the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal-field coupling Δ with the goal of determining the universality class of transitions along the second-order part of the transition line as one approaches the putative location of the multicritical point. We employ extensive Monte Carlo simulations using two different methodologies: (i) a study of the zeros of the energy probability distribution, closely related to the Fisher zeros of the partition function, and (ii) the well-established multicanonical approach employed to study the probability distribution of the crystal-field energy. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram supports previous claims that the transition belongs to the universality class of the four-state Potts model. For positive values of Δ, we observe the presence of strong finite-size effects, indicative of crossover effects due to the proximity of the first-order part of the transition line. Finally, we demonstrate how a combination of cluster and heat-bath updates allows one to equilibrate larger systems, and we demonstrate the potential of this approach for resolving the ambiguities observed in the regime of Δ≳0.

Universality in the two-dimensional dilute Baxter-Wu model

We study the question of universality in the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal field Δ. We employ extensive numerical simulations of two types, providing us with complementary results: Wang-Landau sampling at fixed values of Δ and a parallelized variant of the multicanonical approach performed at constant temperature T. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram indicates that the transition belongs to the universality class of the four-state Potts model. Previous controversies with respect to the nature of the transition are discussed and attributed to the presence of strong finite-size effects, especially as one approaches the pentacritical point of the model.

Finite-size scaling at fixed renormalization-group invariant

Finite-size scaling at fixed renormalization-group invariant is a powerful and flexible technique to analyze Monte Carlo data at a critical point. It consists in fixing a given renormalization-group invariant quantity to a given value, thereby trading its statistical fluctuations with those of a parameter driving the transition. One remarkable feature is the observed significant improvement of statistical accuracy of various quantities, as compared to a standard analysis. We review the method, discussing in detail its implementation, the error analysis, and a previously introduced covariance-based optimization. Comprehensive benchmarks on the Ising model in two and three dimensions show large gains in the statistical accuracy, which are due to cross-correlations between observables. As an application, we compute an accurate estimate of the inverse critical temperature of the improved O(2) ϕ^{4} model on a three-dimensional cubic lattice.

Combining dense and sparse labeling in optical DNA mapping

Optical DNA mapping (ODM) is based on fluorescent labeling, stretching and imaging of single DNA molecules to obtain sequence-specific fluorescence profiles, DNA barcodes. These barcodes can be mapped to theoretical counterparts obtained from DNA reference sequences, which in turn allow for DNA identification in complex samples and for detecting structural changes in individual DNA molecules. There are several types of DNA labeling schemes for ODM and for each labeling type one or several types of match scoring methods are used. By combining the information from multiple labeling schemes one can potentially improve mapping confidence; however, combining match scores from different labeling assays has not been implemented yet. In this study, we introduce two theoretical methods for dealing with analysis of DNA molecules with multiple label types. In our first method, we convert the alignment scores, given as output from the different assays, into p-values using carefully crafted null models. We then combine the p-values for different label types using standard methods to obtain a combined match score and an associated combined p-value. In the second method, we use a block bootstrap approach to check for the uniqueness of a match to a database for all barcodes matching with a combined p-value below a predefined threshold. For obtaining experimental dual-labeled DNA barcodes, we introduce a novel assay where we cut plasmid DNA molecules from bacteria with restriction enzymes and the cut sites serve as sequence-specific markers, which together with barcodes obtained using the established competitive binding labeling method, form a dual-labeled barcode. All experimental data in this study originates from this assay, but we point out that our theoretical framework can be used to combine data from all kinds of available optical DNA mapping assays. We test our multiple labeling frameworks on barcodes from two different plasmids and synthetically generated barcodes (combined competitive-binding- and nick-labeling). It is demonstrated that by simultaneously using the information from all label types, we can substantially increase the significance when we match experimental barcodes to a database consisting of theoretical barcodes for all sequenced plasmids.

Understanding population annealing Monte Carlo simulations

Population annealing is a recent addition to the arsenal of the practitioner in computer simulations in statistical physics and it proves to deal well with systems with complex free-energy landscapes. Above all else, it promises to deliver unrivaled parallel scaling qualities, being suitable for parallel machines of the biggest caliber. Here we study population annealing using as the main example the two-dimensional Ising model, which allows for particularly clean comparisons due to the available exact results and the wealth of published simulational studies employing other approaches. We analyze in depth the accuracy and precision of the method, highlighting its relation to older techniques such as simulated annealing and thermodynamic integration. We introduce intrinsic approaches for the analysis of statistical and systematic errors and provide a detailed picture of the dependence of such errors on the simulation parameters. The results are benchmarked against canonical and parallel tempering simulations.

Phase diagram for the bisected-hexagonal-lattice five-state Potts antiferromagnet

In this paper we study the phase diagram of the five-state Potts antiferromagnet on the bisected-hexagonal lattice. This question is important since Delfino and Tartaglia recently showed that a second-order transition in a five-state Potts antiferromagnet is allowed, and the bisected-hexagonal lattice had emerged as a candidate for such a transition on numerical grounds. By using high-precision Monte Carlo simulations and two complementary analysis methods, we conclude that there is a finite-temperature first-order transition point. This one separates a paramagnetic high-temperature phase, and a low-temperature phase where five phases coexist. This phase transition is very weak in the sense that its latent heat (per edge) is two orders of magnitude smaller than that of other well-known weak first-order phase transitions.

High-resolution Monte Carlo study of the order-parameter distribution of the three-dimensional Ising model

We apply extensive Monte Carlo simulations to study the probability distribution P(m) of the order parameter m for the simple cubic Ising model with periodic boundary condition at the transition point. Sampling is performed with the Wolff cluster flipping algorithm, and histogram reweighting together with finite-size scaling analyses are then used to extract a precise functional form for the probability distribution of the magnetization, P(m), in the thermodynamic limit. This form should serve as a benchmark for other models in the three-dimensional Ising universality class.

Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model

While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.

Efficient simulation of the random-cluster model

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 critical slowing down than the more well-known algorithm due to Swendsen and Wang. Here, we present an efficient implementation of Sweeny's approach for the random-cluster model using recent algorithmic advances in dynamic connectivity algorithms.

Thermal stability of hydrophobic helical oligomers: a lattice simulation study in explicit water

We investigate the thermal stability of helical hydrophobic oligomers using a three-dimensional, water-explicit lattice model and the Wang-Landau Monte Carlo method. The degree of oligomer helicity is controlled by the parameter ε(mm) < 0, which mimics monomer-monomer hydrogen bond interactions leading to the formation of helical turns in atomistic proteins. We vary |ε(mm)| between 0 and 4.5 kcal/mol and therefore investigate systems ranging from flexible homopolymers (i.e., those with no secondary structure) to helical oligomers that are stable over a broad range of temperatures. We find that systems with |ε(mm)| ≤ 2.0 kcal/mol exhibit a broad thermal unfolding transition at high temperature, leading to an ensemble of random coils. In contrast, the structure of conformations involved in a second, low-temperature, transition is strongly dependent on |ε(mm)|. Weakly helical oligomers are observed when |ε(mm)| ≤ 1.0 kcal/mol and exhibit a low-temperature, cold-unfolding-like transition to an ensemble of strongly water-penetrated globular conformations. For higher |ε(mm)| (1.7 kcal/mol ≤ |ε(mm)| ≤ 2.0 kcal/mol), cold unfolding is suppressed, and the low-temperature conformational transition becomes a "crystallization", in which a "molten" helix is transformed into a defect-free helix. The molten helix preserves ≥50% of the helical contacts observed in the "crystal" at a lower temperature. When |ε(mm)| = 4.5 kcal/mol, we find that conformational transitions are largely suppressed within the range of temperatures investigated.

Improvement of Monte Carlo estimates with covariance-optimized finite-size scaling at fixed phenomenological coupling

In the finite-size scaling analysis of Monte Carlo data, instead of computing the observables at fixed Hamiltonian parameters, one may choose to keep a renormalization-group invariant quantity, also called phenomenological coupling, fixed at a given value. Within this scheme of finite-size scaling, we exploit the statistical covariance between the observables in a Monte Carlo simulation in order to reduce the statistical errors of the quantities involved in the computation of the critical exponents. This method is general and does not require additional computational time. This approach is demonstrated in the Ising model in two and three dimensions, where large gain factors in CPU time are obtained.