M.P. Nightingale

Dynamic exponent of the two-dimensional ising model and Monte Carlo computation of the subdominant eigenvalue of the stochastic matrix

We introduce a novel variance-reducing Monte Carlo algorithm for accurate determination of correlation times. We apply this method to two-dimensional Ising systems with sizes up to 15 3 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 correlation times, the dynamic critical exponent z is determined as z › 2.1665s12d. [S0031-9007(96)00379-1]

Critical exponents of two-dimensional Potts and bond percolation models

Critical exponents of the two-dimensional, q-state Potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q. Results for the temperature exponent agree accurately with the conjecture of den Nijs (1979). The magnetic exponent is found to behave in accordance with the conjecture of Nienhuis et al. (1980).

Finite size scaling and critical point exponents of the potts model

The calculation of critical exponents by combining finite size scaling and the transfer matrix technique is proposed and applied to the two-dimensional q-state Potts model. The exact results for q = 2 are very accurately reproduced. For q = 3, our results suggest α = 13and δ = 14. Convergence of our results for q ⩾ 4 is poor but it is suggested that α

Universal Ising dynamics in two dimensions

12and δ #62 14 for q = 4. A preliminary result for the dynamical exponent of the stochastic Ising model is reported.

Finite-Size Interaction Amplitudes and their Universality: Exact, Mean-Field, and Renormalization-Group Results

We explore several dominant eigenvalues of the spectra of Markov matrices governing the dynamics of models in the universality class of the two-dimensional Ising model. By means of a variational approximation, we determine autocorrelation times of progressively rapid relaxation modes. The approximation of one eigenstate, associated with the slowest mode, is employed in a variance-reducing Monte-Carlo method. The resulting correlation times, for which statistical errors exceed the systematic errors associated with the variational approximation, are used for a finite-size scaling analysis which corroborates universality of the dynamic critical exponent z for three distinct Ising models on the square lattice. Tentative, variational results for subdominant states strongly suggest that the amplitudes of the divergent time scales associated with different relaxation modes differ solely by metric factors, setting a single non-universal time scale for each model. A by-product of our analysis is a highly accurate confirmation of static universality.