Energy-decreasing dynamics in mean-field spin models

We perform a statistical analysis of deterministic energy-decreasing algorithms on mean-field spin models with a complex energy landscape, such as the Sine model and the Sherrington-Kirkpatrick model. We specifically address the following question: in the search for low-energy configurations, which is more favorable (and in which sense) - a quick decrease along the gradient (greedy dynamics) or a slow decrease close to the level curves (reluctant dynamics)? Average time and wideness of the attraction basins are introduced for each algorithm, together with an interpolation among the two, and experimental results are presented for different system sizes. We found that while the reluctant algorithm performs better for a fixed number of trials, the two algorithms become basically equivalent for a given elapsed time due to the fact that the greedy algorithm has a shorter relaxation time which scales linearly with the system size compared to a quadratic dependence for the reluctant algorithm.