An efficient, multiple range random walk algorithm to calculate the density of states

We present a new Monte Carlo algorithm that produces results of high accuracy with reduced simulational effort. Independent random walks are performed (concurrently or serially) in different, restricted ranges of energy, and the resultant density of states is modified continuously to produce locally flat histograms. This method permits us to directly access the free energy and entropy, is independent of temperature, and is efficient for the study of both 1st order and 2nd order phase transitions. It should also be useful for the study of complex systems with a rough energy landscape.Comment: 4 pages including 4 ps fig

Flat Histogram Method of Wang-Landau and N-fold Way

We present a method for estimating the density of states of a classical statistical model. The algorithm successfully combines the Wang-Landau flat histogram method with the N-fold way in order to improve efficiency of the original single spin flip version. We test our implementation of the Wang-Landau method with the two-dimensional nearest neighbor Ising model for which we determine the tunneling time and the density of states. Furthermore, we show that our new algorithm performs correctly at right edges of an energy interval over which the density of states is computed. This removes a disadvantage of the original single spin flip Wang-Landau method where results showed systematically higher errors in the density of states at right boundaries. We compare our data with the detailed numerical tests presented in a study by Wang and Swendsen where the original Wang-Landau method was tested against various other methods. Finally, we apply our method to a thin Ising film of size $32\times 32\times 6$ with antisymmetric surface fields. With the density of states obtained from the simulations we calculate canonical averages related to the energy such as internal energy, Gibbs free energy and entropy, but we also sample microcanonical averages during simulations in order to determine canonical averages of the susceptibility, the order parameter and its fourth order cumulant. We compare our results with simulational data obtained from a conventional MC algorithm.Comment: Latex, 19 pages, 12 encapsulated Postscript figure

Nested sampling for Potts models

Nested sampling is a new Monte Carlo method by Skilling [1] intended for general Bayesian computation. Nested sampling provides a robust alternative to annealing-based methods for computing normalizing constants. It can also generate estimates of other quantities such as posterior expectations. The key technical requirement is an ability to draw samples uniformly from the prior subject to a constraint on the likelihood. We provide a demonstration with the Potts model, an undirected graphical model

Markov chain sampling of the O(n)O(n) loop models on the infinite plane

It was recently proposed in https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.043322 [Herdeiro & Doyon Phys.,Rev.,E (2016)] a numerical method showing a precise sampling of the infinite plane 2d critical Ising model for finite lattice subsections. The present note extends the method to a larger class of models, namely the $O(n)$ loop gas models for $n \in (1,2]$. We argue that even though the Gibbs measure is non local, it is factorizable on finite subsections when sufficient information on the loops touching the boundaries is stored. Our results attempt to show that provided an efficient Markov chain mixing algorithm and an improved discrete lattice dilation procedure the planar limit of the $O(n)$ models can be numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the only requirement for the present numerical method to work.Comment: v2: added conclusion section, changes in introduction and appendice

Overcoming the critical slowing down of flat-histogram Monte Carlo simulations: Cluster updates and optimized broad-histogram ensembles

We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform sub-optimally in comparison to an unbiased Markovian random walk in energy space. For the d=1,2,3 Ising model, the mean first-passage time \tau scales with the number of spins N=L^d as \tau \propto N^2L^z. The critical exponent z is found to decrease as the dimensionality d is increased. In the mean-field limit of infinite dimensions we find that z vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by z>0 for finite d can be overcome by two complementary approaches - cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that \tau \propto N^2 up to logarithmic corrections for the d=1 and d=2 Ising model

Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram

We describe an efficient Monte Carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models. The density of states is modified at each step when the energy level is visited to produce a flat histogram. By carefully controlling the modification factor, we allow the density of states to converge to the true value very quickly, even for large systems. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. In this paper, we apply our algorithm to both 1st and 2nd order phase transitions to demonstrate its efficiency and accuracy. We obtained direct simulational estimates for the density of states for two-dimensional ten-state Potts models on lattices up to $200 \times 200$ and Ising models on lattices up to $256 \times 256$. Applying this approach to a 3D $\pm J$ spin glass model we estimate the internal energy and entropy at zero temperature; and, using a two-dimensional random walk in energy and order-parameter space, we obtain the (rough) canonical distribution and energy landscape in order-parameter space. Preliminary data suggest that the glass transition temperature is about 1.2 and that better estimates can be obtained with more extensive application of the method.Comment: 22 pages (figures included

Rapid mixing of Swendsen-Wang dynamics in two dimensions

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature. As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.Comment: Ph.D. thesis, 66 page

Transition Matrix Monte Carlo Reweighting and Dynamics

We study an induced dynamics in the space of energy of single-spin-flip Monte Carlo algorithm. The method gives an efficient reweighting technique. This dynamics is shown to have relaxation times proportional to the specific heat. Thus, it is plausible for a logarithmic factor in the correlation time of the standard 2D Ising local dynamics.Comment: RevTeX, 5 pages, 3 figure