Low-Temperature Long-Time Simulations of Ising Ferromagnets using the Monte Carlo with Absorbing Markov Chains method

The Monte Carlo with Absorbing Markov Chains (MCAMC) method is introduced. This method is a generalization of the rejection-free method known as the $n$-fold way. The MCAMC algorithm is applied to the study of the very low-temperature properties of the lifetime of the metastable state of Ising ferromagnets. This is done both for square-lattice and cubic-lattice nearest-neighbor models. Comparison is made with exact low-temperature predictions, in particular the low-temperature predictions that the metastable lifetime is discontinuous at particular values of the field. This discontinuity for the square lattice is not seen in finite-temperatures studies. For the cubic lattice, it is shown that these `exact predictions' are incorrect near the fields where there are discontinuities. The low-temperature formula must be modified and the corrected low-temperature predictions are not discontinuous in the energy of the nucleating droplet.Comment: Submitted to Computer Physics Communicatinos, for proceedings of the Conference CCP2001, 4 figure

Network Synchronization in a Noisy Environment with Time Delays: Fundamental Limits and Trade-Offs

We study the effects of nonzero time delays in stochastic synchronization problems with linear couplings in an arbitrary network. Using the known exact threshold value from the theory of differential equations with delays, we provide the synchronizability threshold for an arbitrary network. Further, by constructing the scaling theory of the underlying fluctuations, we establish the absolute limit of synchronization efficiency in a noisy environment with uniform time delays, i.e., the minimum attainable value of the width of the synchronization landscape. Our results have also strong implications for optimization and trade-offs in network synchronization with delays.Comment: 3 figure

Diffusion Processes on Power-Law Small-World Networks

We consider diffusion processes on power-law small-world networks in different dimensions. In one dimension, we find a rich phase diagram, with different transient and recurrent phases, including a critical line with continuously varying exponents. The results were obtained using self-consistent perturbation theory and can also be understood in terms of a scaling theory, which provides a general framework for understanding processes on small-world networks with different distributions of long-range links.Comment: 4 pages, 3 figures, added references, modified Fig. 2 with added data (PRL, in press