Monte Carlo computation of correlation times of independent relaxation modes at criticality

We investigate aspects of universality of Glauber critical dynamics in two dimensions. We compute the critical exponent $z$ and numerically corroborate its universality for three different models in the static Ising universality class and for five independent relaxation modes. We also present evidence for universality of amplitude ratios, which shows that, as far as dynamic behavior is concerned, each model in a given universality class is characterized by a single non-universal metric factor which determines the overall time scale. This paper also discusses in detail the variational and projection methods that are used to compute relaxation times with high accuracy

Universal Dynamics of Independent Critical Relaxation Modes

Scaling behavior is studied of several dominant eigenvalues of spectra of Markov matrices and the associated correlation times governing critical slowing down in models in the universality class of the two-dimensional Ising model. A scheme is developed to optimize variational approximants of progressively rapid, independent relaxation modes. These approximants are used to reduce the variance of results obtained by means of an adaptation of a quantum Monte Carlo method to compute eigenvalues subject to errors predominantly of statistical nature. The resulting spectra and correlation times are found to be universal up to a single, non-universal time scale for each model

High-growth firms: introduction to the special section

High-growth firms (HGFs) have attracted considerable attention recently, as academics and policymakers have increasingly recognized the highly skewed nature of many metrics of firm performance. A small number of HGFs drives a disproportionately large amount of job creation, while the average firm has a limited impact on the economy. This article explores the reasons for this increased interest, summarizes the existing literature, and highlights the methodological considerations that constrain and bias research. This special section draws attention to the importance of HGFs for future industrial performance, explores their unusual growth trajectories and strategies, and highlights the lack of persistence of high growth. Consequently, while HGFs are important for understanding the economy and developing public policy, they are unlikely to be useful vehicles for public policy given the difficulties involved in predicting which firms will grow, the lack of persistence in high growth levels, and the complex and often indirect relationship between firm capability, high growth, and macro-economic performance

Transfer-matrix approach to the three-dimensional bond percolation: An application of Novotny's formalism

A transfer-matrix simulation scheme for the three-dimensional (d=3) bond percolation is presented. Our scheme is based on Novotny's transfer-matrix formalism, which enables us to consider arbitrary (integral) number of sites N constituting a unit of the transfer-matrix slice even for d=3. Such an arbitrariness allows us to perform systematic finite-size-scaling analysis of the criticality at the percolation threshold. Diagonalizing the transfer matrix for N =4,5,...,10, we obtain an estimate for the correlation-length critical exponent nu = 0.81(5)

Optimization of ground and excited state wavefunctions and van der Waals clusters

A quantum Monte Carlo method is introduced to optimize excited state trial wavefunctions. The method is applied in a correlation function Monte Carlo calculation to compute ground and excited state energies of bosonic van der Waals clusters of upto seven particles. The calculations are performed using trial wavefunctions with general three-body correlations

Numerical Studies of the Two Dimensional XY Model with Symmetry Breaking Fields

We present results of numerical studies of the two dimensional XY model with four and eight fold symmetry breaking fields. This model has recently been shown to describe hydrogen induced reconstruction on the W(100) surface. Based on mean-field and renormalization group arguments,we first show how the interplay between the anisotropy fields can give rise to different phase transitions in the model. When the fields are compatible with each other there is a continuous phase transition when the fourth order field is varied from negative to positive values. This transition becomes discontinuous at low temperatures. These two regimes are separated by a multicritical point. In the case of competing four and eight fold fields, the first order transition at low temperatures opens up into two Ising transitions. We then use numerical methods to accurately locate the position of the multicritical point, and to verify the nature of the transitions. The different techniques used include Monte Carlo histogram methods combined with finite size scaling analysis, the real space Monte Carlo Renormalization Group method, and the Monte Carlo Transfer Matrix method. Our numerical results are in good agreement with the theoretical arguments.Comment: 29 pages, HU-TFT-94-36, to appear in Phys. Rev. B, Vol 50, November 1, 1994. A LaTeX file with no figure

Corner transfer matrix renormalization group method for two-dimensional self-avoiding walks and other O(n) models

We present an extension of the corner transfer matrix renormalisation group (CTMRG) method to O(n) invariant models, with particular interest in the self-avoiding walk class of models (O(n=0)). The method is illustrated using an interacting self-avoiding walk model. Based on the efficiency and versatility when compared to other available numerical methods, we present CTMRG as the method of choice for two-dimensional self-avoiding walk problems.Comment: 4 pages 7 figures Substantial rewrite of previous version to include calculations of critical points and exponents. Final version accepted for publication in PRE (Rapid Communications

Quantum Speedup by Quantum Annealing

We study the glued-trees problem of Childs et. al. in the adiabatic model of quantum computing and provide an annealing schedule to solve an oracular problem exponentially faster than classically possible. The Hamiltonians involved in the quantum annealing do not suffer from the so-called sign problem. Unlike the typical scenario, our schedule is efficient even though the minimum energy gap of the Hamiltonians is exponentially small in the problem size. We discuss generalizations based on initial-state randomization to avoid some slowdowns in adiabatic quantum computing due to small gaps.Comment: 7 page

The Dynamic Exponent of the Two-Dimensional Ising Model and Monte Carlo Computation of the Sub-Dominant Eigenvalue of the Stochastic Matrix

We introduce a novel variance-reducing Monte Carlo algorithm for accurate determination of autocorrelation times. We apply this method to two-dimensional Ising systems with sizes up to $15 \times 15$, using single-spin flip dynamics, random site selection and transition probabilities according to the heat-bath method. From a finite-size scaling analysis of these autocorrelation times, the dynamical critical exponent $z$ is determined as $z=2.1665$ (12)