A game-theoretic approach to spin models

Monte Carlo algorithms, such as the Metropolis-Hastings one, are commonly used to simulate Ising spin systems. This thesis explores an alternative approach by applying the replicator equation to simulate Ising models, replacing minimisation of the system's energy with the maximisation of the player's strategy. The analysis focuses on an Ising model for which spins vectors can take orientations corresponding to the principal axis of the systems. The replicator equation is modified to take into account interactions between neighbouring spins, both nearest and next-nearest neighbours, as well as external magnetic fields. Results show that both the Metropolis-Hastings and the proposed hybrid algorithm reach similar outcomes at a local level. However, although the classical Monte Carlo-based simulations converge to a more globally stable state, the rate of convergence is significantly slower. The hybrid approach, while much faster, presents a more fragmented state, composed of islands of stable states. The introduction of magnetic fields and nematic interactions greatly impacts the system by enhancing stability at the expenses of a slower convergence rate. The analysis suggests that, while the Metropolis-Hastings algorithm remains a robust method for simulating the Ising model, the replicator dynamics with appropriate modifications, provides a faster and effective tool for understanding stability of complex spin system, by analysing local behaviours.